fimacnv - Intuitionistic Logic Explorer

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Theorem fimacnv 5317

Description: The preimage of the codomain of a mapping is the mapping's domain. (Contributed by FL, 25-Jan-2007.)

**Assertion**
Ref	Expression
fimacnv	⊢ (𝐹:𝐴⟶𝐵 → (^◡𝐹 “ 𝐵) = 𝐴)

**Proof of Theorem fimacnv**
Step	Hyp	Ref	Expression
1		imassrn 4699	. . 3 ⊢ (^◡𝐹 “ 𝐵) ⊆ ran ^◡𝐹
2		dfdm4 4545	. . . 4 ⊢ dom 𝐹 = ran ^◡𝐹
3		fdm 5070	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
4		ssid 3018	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
5	3, 4	syl6eqss 3049	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 ⊆ 𝐴)
6	2, 5	syl5eqssr 3044	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ran ^◡𝐹 ⊆ 𝐴)
7	1, 6	syl5ss 3010	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (^◡𝐹 “ 𝐵) ⊆ 𝐴)
8		imassrn 4699	. . . 4 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹
9		frn 5072	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
10	8, 9	syl5ss 3010	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝐴) ⊆ 𝐵)
11		ffun 5068	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
12	4, 3	syl5sseqr 3048	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 ⊆ dom 𝐹)
13		funimass3 5304	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ 𝐴 ⊆ (^◡𝐹 “ 𝐵)))
14	11, 12, 13	syl2anc 403	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ 𝐴 ⊆ (^◡𝐹 “ 𝐵)))
15	10, 14	mpbid 145	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 ⊆ (^◡𝐹 “ 𝐵))
16	7, 15	eqssd 3016	1 ⊢ (𝐹:𝐴⟶𝐵 → (^◡𝐹 “ 𝐵) = 𝐴)

Colors of variables: wff set class

Syntax hints: → wi 4 ↔ wb 103 = wceq 1284 ⊆ wss 2973 ^◡ccnv 4362 dom cdm 4363 ran crn 4364 “ cima 4366 Fun wfun 4916 ⟶wf 4918

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964

This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930

This theorem is referenced by: fmpt 5340 nn0supp 8340