Theorem onsetreclem1 42448

Description: Lemma for onsetrec 42451. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.)

Hypothesis

Ref	Expression
onsetreclem1.1	|- F = ( x e. _V |-> { U. x , suc U. x } )

Assertion

Ref	Expression
onsetreclem1	|- ( F ` a ) = { U. a , suc U. a }

Distinct variable group:   x, a

Allowed substitution hints:   F( x, a)

Proof of Theorem onsetreclem1

Step	Hyp	Ref	Expression
1		vex 3203	. 2 |- a e. _V
2		unieq 4444	. . . 4 |- ( x = a -> U. x = U. a )
3		suceq 5790	. . . . 5 |- ( U. x = U. a -> suc U. x = suc U. a )
4	2, 3	syl 17	. . . 4 |- ( x = a -> suc U. x = suc U. a )
5	2, 4	preq12d 4276	. . 3 |- ( x = a -> { U. x , suc U. x } = { U. a , suc U. a } )
6		onsetreclem1.1	. . 3 |- F = ( x e. _V |-> { U. x , suc U. x } )
7		prex 4909	. . 3 |- { U. a , suc U. a } e. _V
8	5, 6, 7	fvmpt 6282	. 2 |- ( a e. _V -> ( F ` a ) = { U. a , suc U. a } )
9	1, 8	ax-mp 5	1 |- ( F ` a ) = { U. a , suc U. a }

Syntax hints:   = wceq 1483   e. wcel 1990   _Vcvv 3200   {cpr 4179   U.cuni 4436   |-> cmpt 4729   suc csuc 5725   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-suc 5729  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  onsetreclem2  42449  onsetreclem3  42450