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Theorem fimacnvOLD 7075
Description: Obsolete version of fimacnv 6740 as of 20-Sep-2024. (Contributed by FL, 25-Jan-2007.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
fimacnvOLD (𝐹:𝐴𝐵 → (𝐹𝐵) = 𝐴)

Proof of Theorem fimacnvOLD
StepHypRef Expression
1 imassrn 6069 . . 3 (𝐹𝐵) ⊆ ran 𝐹
2 dfdm4 5892 . . . 4 dom 𝐹 = ran 𝐹
3 fdm 6726 . . . . 5 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
4 ssid 3995 . . . . 5 𝐴𝐴
53, 4eqsstrdi 4027 . . . 4 (𝐹:𝐴𝐵 → dom 𝐹𝐴)
62, 5eqsstrrid 4022 . . 3 (𝐹:𝐴𝐵 → ran 𝐹𝐴)
71, 6sstrid 3984 . 2 (𝐹:𝐴𝐵 → (𝐹𝐵) ⊆ 𝐴)
8 fimass 6738 . . 3 (𝐹:𝐴𝐵 → (𝐹𝐴) ⊆ 𝐵)
9 ffun 6720 . . . 4 (𝐹:𝐴𝐵 → Fun 𝐹)
104, 3sseqtrrid 4026 . . . 4 (𝐹:𝐴𝐵𝐴 ⊆ dom 𝐹)
11 funimass3 7058 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ 𝐵𝐴 ⊆ (𝐹𝐵)))
129, 10, 11syl2anc 582 . . 3 (𝐹:𝐴𝐵 → ((𝐹𝐴) ⊆ 𝐵𝐴 ⊆ (𝐹𝐵)))
138, 12mpbid 231 . 2 (𝐹:𝐴𝐵𝐴 ⊆ (𝐹𝐵))
147, 13eqssd 3990 1 (𝐹:𝐴𝐵 → (𝐹𝐵) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wss 3939  ccnv 5671  dom cdm 5672  ran crn 5673  cima 5675  Fun wfun 6537  wf 6539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551
This theorem is referenced by: (None)
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