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Theorem enmap1lem3 6071
 Description: Lemma for enmap2 6068. Binary relationship condition over W. (Contributed by SF, 3-Mar-2015.)
Hypothesis
Ref Expression
enmap1lem3.1 W = (s (Am G) (r s))
Assertion
Ref Expression
enmap1lem3 (r:A1-1-ontoB → (SWTS = (r T)))
Distinct variable groups:   G,s   s,r   S,s   A,s
Allowed substitution hints:   A(r)   B(s,r)   S(r)   T(s,r)   G(r)   W(s,r)

Proof of Theorem enmap1lem3
StepHypRef Expression
1 breldm 4911 . . . 4 (SWTS dom W)
2 enmap1lem3.1 . . . . . 6 W = (s (Am G) (r s))
32enmap1lem2 6070 . . . . 5 W Fn (Am G)
4 fndm 5182 . . . . 5 (W Fn (Am G) → dom W = (Am G))
53, 4ax-mp 8 . . . 4 dom W = (Am G)
61, 5syl6eleq 2443 . . 3 (SWTS (Am G))
7 fnbrfvb 5358 . . . . . . 7 ((W Fn (Am G) S (Am G)) → ((WS) = TSWT))
83, 7mpan 651 . . . . . 6 (S (Am G) → ((WS) = TSWT))
9 vex 2862 . . . . . . . . 9 r V
10 coexg 4749 . . . . . . . . 9 ((r V S (Am G)) → (r S) V)
119, 10mpan 651 . . . . . . . 8 (S (Am G) → (r S) V)
12 coeq2 4875 . . . . . . . . 9 (s = S → (r s) = (r S))
1312, 2fvmptg 5698 . . . . . . . 8 ((S (Am G) (r S) V) → (WS) = (r S))
1411, 13mpdan 649 . . . . . . 7 (S (Am G) → (WS) = (r S))
1514eqeq1d 2361 . . . . . 6 (S (Am G) → ((WS) = T ↔ (r S) = T))
168, 15bitr3d 246 . . . . 5 (S (Am G) → (SWT ↔ (r S) = T))
1716biimpd 198 . . . 4 (S (Am G) → (SWT → (r S) = T))
186, 17mpcom 32 . . 3 (SWT → (r S) = T)
196, 18jca 518 . 2 (SWT → (S (Am G) (r S) = T))
20 coass 5097 . . . . 5 ((r r) S) = (r (r S))
21 f1ococnv1 5310 . . . . . . 7 (r:A1-1-ontoB → (r r) = ( I A))
2221coeq1d 4878 . . . . . 6 (r:A1-1-ontoB → ((r r) S) = (( I A) S))
23 elmapi 6016 . . . . . . 7 (S (Am G) → S:G–→A)
24 fcoi2 5241 . . . . . . 7 (S:G–→A → (( I A) S) = S)
2523, 24syl 15 . . . . . 6 (S (Am G) → (( I A) S) = S)
2622, 25sylan9eq 2405 . . . . 5 ((r:A1-1-ontoB S (Am G)) → ((r r) S) = S)
2720, 26syl5reqr 2400 . . . 4 ((r:A1-1-ontoB S (Am G)) → S = (r (r S)))
28 coeq2 4875 . . . . 5 ((r S) = T → (r (r S)) = (r T))
2928eqeq2d 2364 . . . 4 ((r S) = T → (S = (r (r S)) ↔ S = (r T)))
3027, 29syl5ibcom 211 . . 3 ((r:A1-1-ontoB S (Am G)) → ((r S) = TS = (r T)))
3130expimpd 586 . 2 (r:A1-1-ontoB → ((S (Am G) (r S) = T) → S = (r T)))
3219, 31syl5 28 1 (r:A1-1-ontoB → (SWTS = (r T)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2859   class class class wbr 4639   ∘ ccom 4721   I cid 4763  ◡ccnv 4771  dom cdm 4772   ↾ cres 4774   Fn wfn 4776  –→wf 4777  –1-1-onto→wf1o 4780   ‘cfv 4781  (class class class)co 5525   ↦ cmpt 5651   ↑m cmap 5999 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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:  enmap1lem4  6072
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