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Theorem fvfullfunlem3 5863
 Description: Lemma for fvfullfun 5864. Part one of the full function definition agrees with the set itself over its domain. (Contributed by SF, 9-Mar-2015.)
Assertion
Ref Expression
fvfullfunlem3 (A dom (( I F) ( ∼ I F)) → ((( I F) ( ∼ I F)) ‘A) = (FA))

Proof of Theorem fvfullfunlem3
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dffun2 5119 . . . 4 (Fun (F dom (( I F) ( ∼ I F))) ↔ xyz((x(F dom (( I F) ( ∼ I F)))y x(F dom (( I F) ( ∼ I F)))z) → y = z))
2 brres 4949 . . . . . . 7 (x(F dom (( I F) ( ∼ I F)))y ↔ (xFy x dom (( I F) ( ∼ I F))))
3 fvfullfunlem1 5861 . . . . . . . . 9 dom (( I F) ( ∼ I F)) = {x ∃!z xFz}
43abeq2i 2460 . . . . . . . 8 (x dom (( I F) ( ∼ I F)) ↔ ∃!z xFz)
54anbi2i 675 . . . . . . 7 ((xFy x dom (( I F) ( ∼ I F))) ↔ (xFy ∃!z xFz))
62, 5bitri 240 . . . . . 6 (x(F dom (( I F) ( ∼ I F)))y ↔ (xFy ∃!z xFz))
7 brres 4949 . . . . . . 7 (x(F dom (( I F) ( ∼ I F)))z ↔ (xFz x dom (( I F) ( ∼ I F))))
8 fvfullfunlem1 5861 . . . . . . . . 9 dom (( I F) ( ∼ I F)) = {x ∃!y xFy}
98abeq2i 2460 . . . . . . . 8 (x dom (( I F) ( ∼ I F)) ↔ ∃!y xFy)
109anbi2i 675 . . . . . . 7 ((xFz x dom (( I F) ( ∼ I F))) ↔ (xFz ∃!y xFy))
117, 10bitri 240 . . . . . 6 (x(F dom (( I F) ( ∼ I F)))z ↔ (xFz ∃!y xFy))
12 tz6.12-1 5344 . . . . . . . . 9 ((xFy ∃!y xFy) → (Fx) = y)
1312adantrl 696 . . . . . . . 8 ((xFy (xFz ∃!y xFy)) → (Fx) = y)
14 tz6.12-1 5344 . . . . . . . . 9 ((xFz ∃!y xFy) → (Fx) = z)
1514adantl 452 . . . . . . . 8 ((xFy (xFz ∃!y xFy)) → (Fx) = z)
1613, 15eqtr3d 2387 . . . . . . 7 ((xFy (xFz ∃!y xFy)) → y = z)
1716adantlr 695 . . . . . 6 (((xFy ∃!z xFz) (xFz ∃!y xFy)) → y = z)
186, 11, 17syl2anb 465 . . . . 5 ((x(F dom (( I F) ( ∼ I F)))y x(F dom (( I F) ( ∼ I F)))z) → y = z)
1918gen2 1547 . . . 4 yz((x(F dom (( I F) ( ∼ I F)))y x(F dom (( I F) ( ∼ I F)))z) → y = z)
201, 19mpgbir 1550 . . 3 Fun (F dom (( I F) ( ∼ I F)))
21 fvfullfunlem2 5862 . . . . 5 (( I F) ( ∼ I F)) F
22 ssdmrn 5099 . . . . . 6 (( I F) ( ∼ I F)) (dom (( I F) ( ∼ I F)) × ran (( I F) ( ∼ I F)))
23 ssv 3291 . . . . . . 7 ran (( I F) ( ∼ I F)) V
24 xpss2 4857 . . . . . . 7 (ran (( I F) ( ∼ I F)) V → (dom (( I F) ( ∼ I F)) × ran (( I F) ( ∼ I F))) (dom (( I F) ( ∼ I F)) × V))
2523, 24ax-mp 5 . . . . . 6 (dom (( I F) ( ∼ I F)) × ran (( I F) ( ∼ I F))) (dom (( I F) ( ∼ I F)) × V)
2622, 25sstri 3281 . . . . 5 (( I F) ( ∼ I F)) (dom (( I F) ( ∼ I F)) × V)
2721, 26ssini 3478 . . . 4 (( I F) ( ∼ I F)) (F ∩ (dom (( I F) ( ∼ I F)) × V))
28 df-res 4788 . . . 4 (F dom (( I F) ( ∼ I F))) = (F ∩ (dom (( I F) ( ∼ I F)) × V))
2927, 28sseqtr4i 3304 . . 3 (( I F) ( ∼ I F)) (F dom (( I F) ( ∼ I F)))
30 funssfv 5343 . . 3 ((Fun (F dom (( I F) ( ∼ I F))) (( I F) ( ∼ I F)) (F dom (( I F) ( ∼ I F))) A dom (( I F) ( ∼ I F))) → ((F dom (( I F) ( ∼ I F))) ‘A) = ((( I F) ( ∼ I F)) ‘A))
3120, 29, 30mp3an12 1267 . 2 (A dom (( I F) ( ∼ I F)) → ((F dom (( I F) ( ∼ I F))) ‘A) = ((( I F) ( ∼ I F)) ‘A))
32 fvres 5342 . 2 (A dom (( I F) ( ∼ I F)) → ((F dom (( I F) ( ∼ I F))) ‘A) = (FA))
3331, 32eqtr3d 2387 1 (A dom (( I F) ( ∼ I F)) → ((( I F) ( ∼ I F)) ‘A) = (FA))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  Vcvv 2859   ∼ ccompl 3205   ∖ cdif 3206   ∩ cin 3208   ⊆ wss 3257   class class class wbr 4639   ∘ ccom 4721   I cid 4763   × cxp 4770  dom cdm 4772  ran crn 4773   ↾ cres 4774  Fun wfun 4775   ‘cfv 4781 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-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fv 4795 This theorem is referenced by:  fvfullfun  5864
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