Metamath Proof Explorer


Theorem tfrlem10

Description: Lemma for transfinite recursion. We define class C by extending recs with one ordered pair. We will assume, falsely, that domain of recs is a member of, and thus not equal to, On . Using this assumption we will prove facts about C that will lead to a contradiction in tfrlem14 , thus showing the domain of recs does in fact equal On . Here we show (under the false assumption) that C is a function extending the domain of recs ( F ) by one. (Contributed by NM, 14-Aug-1994) (Revised by Mario Carneiro, 9-May-2015)

Ref Expression
Hypotheses tfrlem.1 A = f | x On f Fn x y x f y = F f y
tfrlem.3 C = recs F dom recs F F recs F
Assertion tfrlem10 dom recs F On C Fn suc dom recs F

Proof

Step Hyp Ref Expression
1 tfrlem.1 A = f | x On f Fn x y x f y = F f y
2 tfrlem.3 C = recs F dom recs F F recs F
3 fvex F recs F V
4 funsng dom recs F On F recs F V Fun dom recs F F recs F
5 3 4 mpan2 dom recs F On Fun dom recs F F recs F
6 1 tfrlem7 Fun recs F
7 5 6 jctil dom recs F On Fun recs F Fun dom recs F F recs F
8 3 dmsnop dom dom recs F F recs F = dom recs F
9 8 ineq2i dom recs F dom dom recs F F recs F = dom recs F dom recs F
10 1 tfrlem8 Ord dom recs F
11 orddisj Ord dom recs F dom recs F dom recs F =
12 10 11 ax-mp dom recs F dom recs F =
13 9 12 eqtri dom recs F dom dom recs F F recs F =
14 funun Fun recs F Fun dom recs F F recs F dom recs F dom dom recs F F recs F = Fun recs F dom recs F F recs F
15 7 13 14 sylancl dom recs F On Fun recs F dom recs F F recs F
16 8 uneq2i dom recs F dom dom recs F F recs F = dom recs F dom recs F
17 dmun dom recs F dom recs F F recs F = dom recs F dom dom recs F F recs F
18 df-suc suc dom recs F = dom recs F dom recs F
19 16 17 18 3eqtr4i dom recs F dom recs F F recs F = suc dom recs F
20 df-fn recs F dom recs F F recs F Fn suc dom recs F Fun recs F dom recs F F recs F dom recs F dom recs F F recs F = suc dom recs F
21 15 19 20 sylanblrc dom recs F On recs F dom recs F F recs F Fn suc dom recs F
22 2 fneq1i C Fn suc dom recs F recs F dom recs F F recs F Fn suc dom recs F
23 21 22 sylibr dom recs F On C Fn suc dom recs F