acunirnmpt2

Description: Axiom of choice for the union of the range of a mapping to function. (Contributed by Thierry Arnoux, 7-Nov-2019) Remove dependency on ax-reg and ax-inf2 . (Revised by Timothy J. O'Malley, 15-Oct-2026)

	Ref	Expression
Hypotheses	acunirnmpt.0	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
	acunirnmpt.1	⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → 𝐵 ≠ ∅ )
	acunirnmpt2.2	⊢ 𝐶 = ∪ ran ( 𝑗 ∈ 𝐴 ↦ 𝐵 )
	acunirnmpt2.3	⊢ ( 𝑗 = ( 𝑓 ‘ 𝑥 ) → 𝐵 = 𝐷 )
Assertion	acunirnmpt2	⊢ ( 𝜑 → ∃ 𝑓 ( 𝑓 : 𝐶 ⟶ 𝐴 ∧ ∀ 𝑥 ∈ 𝐶 𝑥 ∈ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		acunirnmpt.0	⊢ ( 𝜑 → 𝐴 ∈ 𝑉 )
2		acunirnmpt.1	⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐴 ) → 𝐵 ≠ ∅ )
3		acunirnmpt2.2	⊢ 𝐶 = ∪ ran ( 𝑗 ∈ 𝐴 ↦ 𝐵 )
4		acunirnmpt2.3	⊢ ( 𝑗 = ( 𝑓 ‘ 𝑥 ) → 𝐵 = 𝐷 )
5		mptexg	⊢ ( 𝐴 ∈ 𝑉 → ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) ∈ V )
6		rnexg	⊢ ( ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) ∈ V → ran ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) ∈ V )
7		uniexg	⊢ ( ran ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) ∈ V → ∪ ran ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) ∈ V )
8	1 5 6 7	4syl	⊢ ( 𝜑 → ∪ ran ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) ∈ V )
9	3 8	eqeltrid	⊢ ( 𝜑 → 𝐶 ∈ V )
10		eqid	⊢ ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑗 ∈ 𝐴 ↦ 𝐵 )
11		simplr	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) ∧ 𝑦 ∈ ran ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) ) ∧ 𝑥 ∈ 𝑦 ) → 𝑦 ∈ ran ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) )
12	10 11	elrnmpt2d	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) ∧ 𝑦 ∈ ran ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) ) ∧ 𝑥 ∈ 𝑦 ) → ∃ 𝑗 ∈ 𝐴 𝑦 = 𝐵 )
13		nfv	⊢ Ⅎ 𝑗 ( 𝜑 ∧ 𝑥 ∈ 𝐶 )
14		nfmpt1	⊢ Ⅎ 𝑗 ( 𝑗 ∈ 𝐴 ↦ 𝐵 )
15	14	nfrn	⊢ Ⅎ 𝑗 ran ( 𝑗 ∈ 𝐴 ↦ 𝐵 )
16	15	nfcri	⊢ Ⅎ 𝑗 𝑦 ∈ ran ( 𝑗 ∈ 𝐴 ↦ 𝐵 )
17	13 16	nfan	⊢ Ⅎ 𝑗 ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) ∧ 𝑦 ∈ ran ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) )
18		nfv	⊢ Ⅎ 𝑗 𝑥 ∈ 𝑦
19	17 18	nfan	⊢ Ⅎ 𝑗 ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) ∧ 𝑦 ∈ ran ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) ) ∧ 𝑥 ∈ 𝑦 )
20		simpllr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) ∧ 𝑦 ∈ ran ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) ) ∧ 𝑥 ∈ 𝑦 ) ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑦 = 𝐵 ) → 𝑥 ∈ 𝑦 )
21		simpr	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) ∧ 𝑦 ∈ ran ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) ) ∧ 𝑥 ∈ 𝑦 ) ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑦 = 𝐵 ) → 𝑦 = 𝐵 )
22	20 21	eleqtrd	⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) ∧ 𝑦 ∈ ran ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) ) ∧ 𝑥 ∈ 𝑦 ) ∧ 𝑗 ∈ 𝐴 ) ∧ 𝑦 = 𝐵 ) → 𝑥 ∈ 𝐵 )
23	22	exp31	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) ∧ 𝑦 ∈ ran ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) ) ∧ 𝑥 ∈ 𝑦 ) → ( 𝑗 ∈ 𝐴 → ( 𝑦 = 𝐵 → 𝑥 ∈ 𝐵 ) ) )
24	19 23	reximdai	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) ∧ 𝑦 ∈ ran ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) ) ∧ 𝑥 ∈ 𝑦 ) → ( ∃ 𝑗 ∈ 𝐴 𝑦 = 𝐵 → ∃ 𝑗 ∈ 𝐴 𝑥 ∈ 𝐵 ) )
25	12 24	mpd	⊢ ( ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) ∧ 𝑦 ∈ ran ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) ) ∧ 𝑥 ∈ 𝑦 ) → ∃ 𝑗 ∈ 𝐴 𝑥 ∈ 𝐵 )
26	3	eleq2i	⊢ ( 𝑥 ∈ 𝐶 ↔ 𝑥 ∈ ∪ ran ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) )
27		eluni2	⊢ ( 𝑥 ∈ ∪ ran ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) ↔ ∃ 𝑦 ∈ ran ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) 𝑥 ∈ 𝑦 )
28	26 27	sylbb	⊢ ( 𝑥 ∈ 𝐶 → ∃ 𝑦 ∈ ran ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) 𝑥 ∈ 𝑦 )
29	28	adantl	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) → ∃ 𝑦 ∈ ran ( 𝑗 ∈ 𝐴 ↦ 𝐵 ) 𝑥 ∈ 𝑦 )
30	25 29	r19.29a	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐶 ) → ∃ 𝑗 ∈ 𝐴 𝑥 ∈ 𝐵 )
31	30	ralrimiva	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐶 ∃ 𝑗 ∈ 𝐴 𝑥 ∈ 𝐵 )
32	4	eleq2d	⊢ ( 𝑗 = ( 𝑓 ‘ 𝑥 ) → ( 𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝐷 ) )
33	32	ac6	⊢ ( ( 𝐶 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ ∀ 𝑥 ∈ 𝐶 ∃ 𝑗 ∈ 𝐴 𝑥 ∈ 𝐵 ) → ∃ 𝑓 ( 𝑓 : 𝐶 ⟶ 𝐴 ∧ ∀ 𝑥 ∈ 𝐶 𝑥 ∈ 𝐷 ) )
34	9 1 31 33	syl3anc	⊢ ( 𝜑 → ∃ 𝑓 ( 𝑓 : 𝐶 ⟶ 𝐴 ∧ ∀ 𝑥 ∈ 𝐶 𝑥 ∈ 𝐷 ) )

Metamath Proof Explorer

Theorem acunirnmpt2

Proof