f1orescnv - Metamath Proof Explorer

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Theorem f1orescnv 6065

Description: The converse of a one-to-one-onto restricted function. (Contributed by Paul Chapman, 21-Apr-2008.)

**Assertion**
Ref	Expression
f1orescnv	⊢ ((Fun ^◡𝐹 ∧ (𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃) → (^◡𝐹 ↾ 𝑃):𝑃–1-1-onto→𝑅)

**Proof of Theorem f1orescnv**
Step	Hyp	Ref	Expression
1		f1ocnv 6062	. . 3 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 → ^◡(𝐹 ↾ 𝑅):𝑃–1-1-onto→𝑅)
2	1	adantl 481	. 2 ⊢ ((Fun ^◡𝐹 ∧ (𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃) → ^◡(𝐹 ↾ 𝑅):𝑃–1-1-onto→𝑅)
3		funcnvres 5881	. . . 4 ⊢ (Fun ^◡𝐹 → ^◡(𝐹 ↾ 𝑅) = (^◡𝐹 ↾ (𝐹 “ 𝑅)))
4		df-ima 5051	. . . . . 6 ⊢ (𝐹 “ 𝑅) = ran (𝐹 ↾ 𝑅)
5		dff1o5 6059	. . . . . . 7 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 ↔ ((𝐹 ↾ 𝑅):𝑅–1-1→𝑃 ∧ ran (𝐹 ↾ 𝑅) = 𝑃))
6	5	simprbi 479	. . . . . 6 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 → ran (𝐹 ↾ 𝑅) = 𝑃)
7	4, 6	syl5eq 2656	. . . . 5 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 → (𝐹 “ 𝑅) = 𝑃)
8	7	reseq2d 5317	. . . 4 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 → (^◡𝐹 ↾ (𝐹 “ 𝑅)) = (^◡𝐹 ↾ 𝑃))
9	3, 8	sylan9eq 2664	. . 3 ⊢ ((Fun ^◡𝐹 ∧ (𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃) → ^◡(𝐹 ↾ 𝑅) = (^◡𝐹 ↾ 𝑃))
10		f1oeq1 6040	. . 3 ⊢ (^◡(𝐹 ↾ 𝑅) = (^◡𝐹 ↾ 𝑃) → (^◡(𝐹 ↾ 𝑅):𝑃–1-1-onto→𝑅 ↔ (^◡𝐹 ↾ 𝑃):𝑃–1-1-onto→𝑅))
11	9, 10	syl 17	. 2 ⊢ ((Fun ^◡𝐹 ∧ (𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃) → (^◡(𝐹 ↾ 𝑅):𝑃–1-1-onto→𝑅 ↔ (^◡𝐹 ↾ 𝑃):𝑃–1-1-onto→𝑅))
12	2, 11	mpbid 221	1 ⊢ ((Fun ^◡𝐹 ∧ (𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃) → (^◡𝐹 ↾ 𝑃):𝑃–1-1-onto→𝑅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ^◡ccnv 5037 ran crn 5039 ↾ cres 5040 “ cima 5041 Fun wfun 5798 –1-1→wf1 5801 –1-1-onto→wf1o 5803

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811

This theorem is referenced by: f1oresrab 6302 relogf1o 24117

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