Theorem f1mptrn 29435

Description: Express injection for a mapping operation. (Contributed by Thierry Arnoux, 3-May-2020.)

Hypotheses

Ref	Expression
f1mptrn.1	|- ( ( ph /\ x e. A ) -> B e. C )
f1mptrn.2	|- ( ( ph /\ y e. C ) -> E! x e. A y = B )

Assertion

Ref	Expression
f1mptrn	|- ( ph -> Fun `' ( x e. A |-> B ) )

Distinct variable groups:   x, A, y   y, B   x, C, y   ph, x, y

Allowed substitution hint:   B( x)

Proof of Theorem f1mptrn

Step	Hyp	Ref	Expression
1		f1mptrn.1	. . . 4 |- ( ( ph /\ x e. A ) -> B e. C )
2	1	ralrimiva 2966	. . 3 |- ( ph -> A. x e. A B e. C )
3		f1mptrn.2	. . . 4 |- ( ( ph /\ y e. C ) -> E! x e. A y = B )
4	3	ralrimiva 2966	. . 3 |- ( ph -> A. y e. C E! x e. A y = B )
5	2, 4	jca 554	. 2 |- ( ph -> ( A. x e. A B e. C /\ A. y e. C E! x e. A y = B ) )
6		eqid 2622	. . . 4 |- ( x e. A |-> B ) = ( x e. A |-> B )
7	6	f1ompt 6382	. . 3 |- ( ( x e. A |-> B ) : A -1-1-onto-> C <-> ( A. x e. A B e. C /\ A. y e. C E! x e. A y = B ) )
8		dff1o2 6142	. . . 4 |- ( ( x e. A |-> B ) : A -1-1-onto-> C <-> ( ( x e. A |-> B ) Fn A /\ Fun `' ( x e. A |-> B ) /\ ran ( x e. A |-> B ) = C ) )
9	8	simp2bi 1077	. . 3 |- ( ( x e. A |-> B ) : A -1-1-onto-> C -> Fun `' ( x e. A |-> B ) )
10	7, 9	sylbir 225	. 2 |- ( ( A. x e. A B e. C /\ A. y e. C E! x e. A y = B ) -> Fun `' ( x e. A |-> B ) )
11	5, 10	syl 17	1 |- ( ph -> Fun `' ( x e. A |-> B ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E!wreu 2914   |-> cmpt 4729   `'ccnv 5113   ran crn 5115   Fun wfun 5882   Fn wfn 5883   -1-1-onto->wf1o 5887

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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  esum2dlem  30154