fimacnv - Intuitionistic Logic Explorer

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Theorem fimacnv 5784

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 5093	. . 3 ⊢ (^◡𝐹 “ 𝐵) ⊆ ran ^◡𝐹
2		dfdm4 4929	. . . 4 ⊢ dom 𝐹 = ran ^◡𝐹
3		fdm 5495	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
4		ssid 3248	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
5	3, 4	eqsstrdi 3280	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 ⊆ 𝐴)
6	2, 5	eqsstrrid 3275	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ran ^◡𝐹 ⊆ 𝐴)
7	1, 6	sstrid 3239	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (^◡𝐹 “ 𝐵) ⊆ 𝐴)
8		imassrn 5093	. . . 4 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹
9		frn 5498	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
10	8, 9	sstrid 3239	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝐴) ⊆ 𝐵)
11		ffun 5492	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
12	4, 3	sseqtrrid 3279	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 ⊆ dom 𝐹)
13		funimass3 5772	. . . 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 3245	1 ⊢ (𝐹:𝐴⟶𝐵 → (^◡𝐹 “ 𝐵) = 𝐴)

Colors of variables: wff set class

Syntax hints: → wi 4 ↔ wb 105 = wceq 1398 ⊆ wss 3201 ^◡ccnv 4730 dom cdm 4731 ran crn 4732 “ cima 4734 Fun wfun 5327 ⟶wf 5329

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 2205 ax-ext 2213 ax-sep 4212 ax-pow 4270 ax-pr 4305

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ral 2516 df-rex 2517 df-v 2805 df-sbc 3033 df-un 3205 df-in 3207 df-ss 3214 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-br 4094 df-opab 4156 df-id 4396 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-fv 5341

This theorem is referenced by: fmpt 5805 fsuppeq 6425 fsuppeqg 6426 nn0supp 9515 cnclima 15034