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Mirrors > Home > MPE Home > Th. List > fvresex | Unicode version |
Description: Existence of the class of values of a restricted class. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 11-Sep-2015.) |
Ref | Expression |
---|---|
fvresex.1 |
Ref | Expression |
---|---|
fvresex |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssv 3523 | . . . . . . . 8 | |
2 | resmpt 5328 | . . . . . . . 8 | |
3 | 1, 2 | ax-mp 5 | . . . . . . 7 |
4 | 3 | fveq1i 5872 | . . . . . 6 |
5 | vex 3112 | . . . . . . . 8 | |
6 | fveq2 5871 | . . . . . . . . 9 | |
7 | eqid 2457 | . . . . . . . . 9 | |
8 | fvex 5881 | . . . . . . . . 9 | |
9 | 6, 7, 8 | fvmpt 5956 | . . . . . . . 8 |
10 | 5, 9 | ax-mp 5 | . . . . . . 7 |
11 | fveqres 5905 | . . . . . . 7 | |
12 | 10, 11 | ax-mp 5 | . . . . . 6 |
13 | 4, 12 | eqtr3i 2488 | . . . . 5 |
14 | 13 | eqeq2i 2475 | . . . 4 |
15 | 14 | exbii 1667 | . . 3 |
16 | 15 | abbii 2591 | . 2 |
17 | fvresex.1 | . . . 4 | |
18 | 17 | mptex 6143 | . . 3 |
19 | 18 | fvclex 6772 | . 2 |
20 | 16, 19 | eqeltrri 2542 | 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1395 E. wex 1612
e. wcel 1818 { cab 2442 cvv 3109
C_ wss 3475 e. cmpt 4510 |` cres 5006
` cfv 5593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-8 1820 ax-9 1822 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 ax-rep 4563 ax-sep 4573 ax-nul 4581 ax-pow 4630 ax-pr 4691 ax-un 6592 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 975 df-tru 1398 df-ex 1613 df-nf 1617 df-sb 1740 df-eu 2286 df-mo 2287 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-ral 2812 df-rex 2813 df-reu 2814 df-rab 2816 df-v 3111 df-sbc 3328 df-csb 3435 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-nul 3785 df-if 3942 df-pw 4014 df-sn 4030 df-pr 4032 df-op 4036 df-uni 4250 df-iun 4332 df-br 4453 df-opab 4511 df-mpt 4512 df-id 4800 df-xp 5010 df-rel 5011 df-cnv 5012 df-co 5013 df-dm 5014 df-rn 5015 df-res 5016 df-ima 5017 df-iota 5556 df-fun 5595 df-fn 5596 df-f 5597 df-f1 5598 df-fo 5599 df-f1o 5600 df-fv 5601 |
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