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Theorem fimacnvOLD 7066
Description: Obsolete version of fimacnv 6733 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 6064 . . 3 (𝐹𝐵) ⊆ ran 𝐹
2 dfdm4 5889 . . . 4 dom 𝐹 = ran 𝐹
3 fdm 6720 . . . . 5 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
4 ssid 3999 . . . . 5 𝐴𝐴
53, 4eqsstrdi 4031 . . . 4 (𝐹:𝐴𝐵 → dom 𝐹𝐴)
62, 5eqsstrrid 4026 . . 3 (𝐹:𝐴𝐵 → ran 𝐹𝐴)
71, 6sstrid 3988 . 2 (𝐹:𝐴𝐵 → (𝐹𝐵) ⊆ 𝐴)
8 fimass 6732 . . 3 (𝐹:𝐴𝐵 → (𝐹𝐴) ⊆ 𝐵)
9 ffun 6714 . . . 4 (𝐹:𝐴𝐵 → Fun 𝐹)
104, 3sseqtrrid 4030 . . . 4 (𝐹:𝐴𝐵𝐴 ⊆ dom 𝐹)
11 funimass3 7049 . . . 4 ((Fun 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ 𝐵𝐴 ⊆ (𝐹𝐵)))
129, 10, 11syl2anc 583 . . 3 (𝐹:𝐴𝐵 → ((𝐹𝐴) ⊆ 𝐵𝐴 ⊆ (𝐹𝐵)))
138, 12mpbid 231 . 2 (𝐹:𝐴𝐵𝐴 ⊆ (𝐹𝐵))
147, 13eqssd 3994 1 (𝐹:𝐴𝐵 → (𝐹𝐵) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wss 3943  ccnv 5668  dom cdm 5669  ran crn 5670  cima 5672  Fun wfun 6531  wf 6533
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545
This theorem is referenced by: (None)
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