fimacnv - Intuitionistic Logic Explorer

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Theorem fimacnv 5776

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 5087	. . 3 ⊢ (^◡𝐹 “ 𝐵) ⊆ ran ^◡𝐹
2		dfdm4 4923	. . . 4 ⊢ dom 𝐹 = ran ^◡𝐹
3		fdm 5488	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
4		ssid 3247	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
5	3, 4	eqsstrdi 3279	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 ⊆ 𝐴)
6	2, 5	eqsstrrid 3274	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → ran ^◡𝐹 ⊆ 𝐴)
7	1, 6	sstrid 3238	. 2 ⊢ (𝐹:𝐴⟶𝐵 → (^◡𝐹 “ 𝐵) ⊆ 𝐴)
8		imassrn 5087	. . . 4 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹
9		frn 5491	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵)
10	8, 9	sstrid 3238	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 “ 𝐴) ⊆ 𝐵)
11		ffun 5485	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹)
12	4, 3	sseqtrrid 3278	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 ⊆ dom 𝐹)
13		funimass3 5763	. . . 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 3244	1 ⊢ (𝐹:𝐴⟶𝐵 → (^◡𝐹 “ 𝐵) = 𝐴)

Colors of variables: wff set class

Syntax hints: → wi 4 ↔ wb 105 = wceq 1397 ⊆ wss 3200 ^◡ccnv 4724 dom cdm 4725 ran crn 4726 “ cima 4728 Fun wfun 5320 ⟶wf 5322

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299

This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ral 2515 df-rex 2516 df-v 2804 df-sbc 3032 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-br 4089 df-opab 4151 df-id 4390 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-fv 5334

This theorem is referenced by: fmpt 5797 nn0supp 9453 cnclima 14946