f1orescnv - Metamath Proof Explorer

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Theorem f1orescnv 6877

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 6874	. . 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 6656	. . . 4 ⊢ (Fun ^◡𝐹 → ^◡(𝐹 ↾ 𝑅) = (^◡𝐹 ↾ (𝐹 “ 𝑅)))
4		df-ima 5713	. . . . . 6 ⊢ (𝐹 “ 𝑅) = ran (𝐹 ↾ 𝑅)
5		dff1o5 6871	. . . . . . 7 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 ↔ ((𝐹 ↾ 𝑅):𝑅–1-1→𝑃 ∧ ran (𝐹 ↾ 𝑅) = 𝑃))
6	5	simprbi 496	. . . . . 6 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 → ran (𝐹 ↾ 𝑅) = 𝑃)
7	4, 6	eqtrid 2792	. . . . 5 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 → (𝐹 “ 𝑅) = 𝑃)
8	7	reseq2d 6009	. . . 4 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 → (^◡𝐹 ↾ (𝐹 “ 𝑅)) = (^◡𝐹 ↾ 𝑃))
9	3, 8	sylan9eq 2800	. . 3 ⊢ ((Fun ^◡𝐹 ∧ (𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃) → ^◡(𝐹 ↾ 𝑅) = (^◡𝐹 ↾ 𝑃))
10	9	f1oeq1d 6857	. 2 ⊢ ((Fun ^◡𝐹 ∧ (𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃) → (^◡(𝐹 ↾ 𝑅):𝑃–1-1-onto→𝑅 ↔ (^◡𝐹 ↾ 𝑃):𝑃–1-1-onto→𝑅))
11	2, 10	mpbid 232	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 ∧ wa 395 = wceq 1537 ^◡ccnv 5699 ran crn 5701 ↾ cres 5702 “ cima 5703 Fun wfun 6567 –1-1→wf1 6570 –1-1-onto→wf1o 6572

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5167 df-opab 5229 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580

This theorem is referenced by: f1oresrab 7161 relogf1o 26626