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:𝐴⟶𝐵 → (◡𝐹 “ 𝐵) = 𝐴)

Proof of Theorem fimacnvOLD

Step	Hyp	Ref	Expression
1		imassrn 6069	. . 3 ⊢ (◡𝐹 “ 𝐵) ⊆ ran ◡𝐹
2		dfdm4 5892	. . . 4 ⊢ dom 𝐹 = ran ◡𝐹
3		fdm 6726	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
4		ssid 3995	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
5	3, 4	eqsstrdi 4027	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 ⊆ 𝐴)
6	2, 5	eqsstrrid 4022	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ran ◡𝐹 ⊆ 𝐴)
7	1, 6	sstrid 3984	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (◡𝐹 “ 𝐵) ⊆ 𝐴)
8		fimass 6738	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝐴) ⊆ 𝐵)
9		ffun 6720	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
10	4, 3	sseqtrrid 4026	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 ⊆ dom 𝐹)
11		funimass3 7058	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ 𝐵)))
12	9, 10, 11	syl2anc 582	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ 𝐴 ⊆ (◡𝐹 “ 𝐵)))
13	8, 12	mpbid 231	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 ⊆ (◡𝐹 “ 𝐵))
14	7, 13	eqssd 3990	1 ⊢ (𝐹:𝐴⟶𝐵 → (◡𝐹 “ 𝐵) = 𝐴)

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)