fimacnv - Intuitionistic Logic Explorer

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Theorem fimacnv 5549

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 4892	. . 3 ⊢ (^◡𝐹 “ 𝐵) ⊆ ran ^◡𝐹
2		dfdm4 4731	. . . 4 ⊢ dom 𝐹 = ran ^◡𝐹
3		fdm 5278	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
4		ssid 3117	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
5	3, 4	eqsstrdi 3149	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 ⊆ 𝐴)
6	2, 5	eqsstrrid 3144	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ran ^◡𝐹 ⊆ 𝐴)
7	1, 6	sstrid 3108	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (^◡𝐹 “ 𝐵) ⊆ 𝐴)
8		imassrn 4892	. . . 4 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹
9		frn 5281	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
10	8, 9	sstrid 3108	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝐴) ⊆ 𝐵)
11		ffun 5275	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
12	4, 3	sseqtrrid 3148	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 ⊆ dom 𝐹)
13		funimass3 5536	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ 𝐴 ⊆ (^◡𝐹 “ 𝐵)))
14	11, 12, 13	syl2anc 408	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ 𝐴 ⊆ (^◡𝐹 “ 𝐵)))
15	10, 14	mpbid 146	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 ⊆ (^◡𝐹 “ 𝐵))
16	7, 15	eqssd 3114	1 ⊢ (𝐹:𝐴⟶𝐵 → (^◡𝐹 “ 𝐵) = 𝐴)

Colors of variables: wff set class

Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 ⊆ wss 3071 ^◡ccnv 4538 dom cdm 4539 ran crn 4540 “ cima 4542 Fun wfun 5117 ⟶wf 5119

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131

This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131

This theorem is referenced by: fmpt 5570 nn0supp 9029 cnclima 12392