Theorem enmap1lem4 6073

Description: Lemma for enmap2 6069. The converse of W is a function. (Contributed by SF, 3-Mar-2015.)

Hypothesis

Ref	Expression
enmap1lem4.1	⊢ W = (s ∈ (A ↑m G) ↦ (r ∘ s))

Assertion

Ref	Expression
enmap1lem4	⊢ (r:A–1-1-onto→B → Fun ◡W)

Distinct variable groups:   G,s   s,r   A,s

Allowed substitution hints:   A(r)   B(s,r)   G(r)   W(s,r)

Proof of Theorem enmap1lem4

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		enmap1lem4.1	. . . . . . 7 ⊢ W = (s ∈ (A ↑m G) ↦ (r ∘ s))
2	1	enmap1lem3 6072	. . . . . 6 ⊢ (r:A–1-1-onto→B → (yWx → y = (◡r ∘ x)))
3	1	enmap1lem3 6072	. . . . . 6 ⊢ (r:A–1-1-onto→B → (zWx → z = (◡r ∘ x)))
4	2, 3	anim12d 546	. . . . 5 ⊢ (r:A–1-1-onto→B → ((yWx ∧ zWx) → (y = (◡r ∘ x) ∧ z = (◡r ∘ x))))
5		eqtr3 2372	. . . . 5 ⊢ ((y = (◡r ∘ x) ∧ z = (◡r ∘ x)) → y = z)
6	4, 5	syl6 29	. . . 4 ⊢ (r:A–1-1-onto→B → ((yWx ∧ zWx) → y = z))
7	6	alrimiv 1631	. . 3 ⊢ (r:A–1-1-onto→B → ∀z((yWx ∧ zWx) → y = z))
8	7	alrimivv 1632	. 2 ⊢ (r:A–1-1-onto→B → ∀x∀y∀z((yWx ∧ zWx) → y = z))
9		dffun2 5120	. . 3 ⊢ (Fun ◡W ↔ ∀x∀y∀z((x◡Wy ∧ x◡Wz) → y = z))
10		brcnv 4893	. . . . . . 7 ⊢ (x◡Wy ↔ yWx)
11		brcnv 4893	. . . . . . 7 ⊢ (x◡Wz ↔ zWx)
12	10, 11	anbi12i 678	. . . . . 6 ⊢ ((x◡Wy ∧ x◡Wz) ↔ (yWx ∧ zWx))
13	12	imbi1i 315	. . . . 5 ⊢ (((x◡Wy ∧ x◡Wz) → y = z) ↔ ((yWx ∧ zWx) → y = z))
14	13	albii 1566	. . . 4 ⊢ (∀z((x◡Wy ∧ x◡Wz) → y = z) ↔ ∀z((yWx ∧ zWx) → y = z))
15	14	2albii 1567	. . 3 ⊢ (∀x∀y∀z((x◡Wy ∧ x◡Wz) → y = z) ↔ ∀x∀y∀z((yWx ∧ zWx) → y = z))
16	9, 15	bitri 240	. 2 ⊢ (Fun ◡W ↔ ∀x∀y∀z((yWx ∧ zWx) → y = z))
17	8, 16	sylibr 203	1 ⊢ (r:A–1-1-onto→B → Fun ◡W)

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540   = wceq 1642   class class class wbr 4640   ∘ ccom 4722  ^◡ccnv 4772  Fun wfun 4776  –1-1-onto→wf1o 4781  (class class class)co 5526   ↦ cmpt 5652   ↑_m cmap 6000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-map 6002

This theorem is referenced by:  enmap1  6075