fimacnv - Intuitionistic Logic Explorer

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Theorem fimacnv 5806

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 5112	. . 3 ⊢ (^◡𝐹 “ 𝐵) ⊆ ran ^◡𝐹
2		dfdm4 4948	. . . 4 ⊢ dom 𝐹 = ran ^◡𝐹
3		fdm 5514	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
4		ssid 3258	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
5	3, 4	eqsstrdi 3290	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 ⊆ 𝐴)
6	2, 5	eqsstrrid 3285	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ran ^◡𝐹 ⊆ 𝐴)
7	1, 6	sstrid 3249	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (^◡𝐹 “ 𝐵) ⊆ 𝐴)
8		imassrn 5112	. . . 4 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹
9		frn 5517	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
10	8, 9	sstrid 3249	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝐴) ⊆ 𝐵)
11		ffun 5511	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
12	4, 3	sseqtrrid 3289	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 ⊆ dom 𝐹)
13		funimass3 5794	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ 𝐴 ⊆ (^◡𝐹 “ 𝐵)))
14	11, 12, 13	syl2anc 411	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ 𝐴 ⊆ (^◡𝐹 “ 𝐵)))
15	10, 14	mpbid 147	. 2 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 ⊆ (^◡𝐹 “ 𝐵))
16	7, 15	eqssd 3255	1 ⊢ (𝐹:𝐴⟶𝐵 → (^◡𝐹 “ 𝐵) = 𝐴)

Colors of variables: wff set class

Syntax hints: → wi 4 ↔ wb 105 = wceq 1398 ⊆ wss 3211 ^◡ccnv 4748 dom cdm 4749 ran crn 4750 “ cima 4752 Fun wfun 5346 ⟶wf 5348

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-14 2206 ax-ext 2214 ax-sep 4228 ax-pow 4287 ax-pr 4322

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ral 2525 df-rex 2526 df-v 2815 df-sbc 3043 df-un 3215 df-in 3217 df-ss 3224 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-br 4110 df-opab 4172 df-id 4414 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-fv 5360

This theorem is referenced by: fmpt 5827 fsuppeq 6447 fsuppeqg 6448 nn0supp 9552 cnclima 15088