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Mirrors > Home > NFE Home > Th. List > enmap2lem4 | GIF version |
Description: Lemma for enmap2 6068. The converse of W is a function. (Contributed by SF, 26-Feb-2015.) |
Ref | Expression |
---|---|
enmap2lem4.1 | ⊢ W = (s ∈ (G ↑m a) ↦ (s ∘ ◡r)) |
Ref | Expression |
---|---|
enmap2lem4 | ⊢ (r:a–1-1-onto→b → Fun ◡W) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | enmap2lem4.1 | . . . . . . 7 ⊢ W = (s ∈ (G ↑m a) ↦ (s ∘ ◡r)) | |
2 | 1 | enmap2lem3 6065 | . . . . . 6 ⊢ (r:a–1-1-onto→b → (yWx → y = (x ∘ r))) |
3 | 1 | enmap2lem3 6065 | . . . . . 6 ⊢ (r:a–1-1-onto→b → (zWx → z = (x ∘ r))) |
4 | 2, 3 | anim12d 546 | . . . . 5 ⊢ (r:a–1-1-onto→b → ((yWx ∧ zWx) → (y = (x ∘ r) ∧ z = (x ∘ r)))) |
5 | eqtr3 2372 | . . . . 5 ⊢ ((y = (x ∘ r) ∧ z = (x ∘ r)) → 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 5119 | . . 3 ⊢ (Fun ◡W ↔ ∀x∀y∀z((x◡Wy ∧ x◡Wz) → y = z)) | |
10 | brcnv 4892 | . . . . . . 7 ⊢ (x◡Wy ↔ yWx) | |
11 | brcnv 4892 | . . . . . . 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) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∀wal 1540 = wceq 1642 class class class wbr 4639 ∘ ccom 4721 ◡ccnv 4771 Fun wfun 4775 –1-1-onto→wf1o 4780 (class class class)co 5525 ↦ cmpt 5651 ↑m cmap 5999 |
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 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
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 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-1st 4723 df-swap 4724 df-sset 4725 df-co 4726 df-ima 4727 df-si 4728 df-id 4767 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-res 4788 df-fun 4789 df-fn 4790 df-f 4791 df-f1 4792 df-fo 4793 df-f1o 4794 df-fv 4795 df-2nd 4797 df-ov 5526 df-oprab 5528 df-mpt 5652 df-mpt2 5654 df-txp 5736 df-ins2 5750 df-ins3 5752 df-image 5754 df-ins4 5756 df-si3 5758 df-funs 5760 df-map 6001 |
This theorem is referenced by: enmap2 6068 |
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