f1orescnv - Metamath Proof Explorer

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Theorem f1orescnv 6731

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 6728	. . 3 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 → ^◡(𝐹 ↾ 𝑅):𝑃–1-1-onto→𝑅)
2	1	adantl 482	. 2 ⊢ ((Fun ^◡𝐹 ∧ (𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃) → ^◡(𝐹 ↾ 𝑅):𝑃–1-1-onto→𝑅)
3		funcnvres 6512	. . . 4 ⊢ (Fun ^◡𝐹 → ^◡(𝐹 ↾ 𝑅) = (^◡𝐹 ↾ (𝐹 “ 𝑅)))
4		df-ima 5602	. . . . . 6 ⊢ (𝐹 “ 𝑅) = ran (𝐹 ↾ 𝑅)
5		dff1o5 6725	. . . . . . 7 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 ↔ ((𝐹 ↾ 𝑅):𝑅–1-1→𝑃 ∧ ran (𝐹 ↾ 𝑅) = 𝑃))
6	5	simprbi 497	. . . . . 6 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 → ran (𝐹 ↾ 𝑅) = 𝑃)
7	4, 6	eqtrid 2790	. . . . 5 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 → (𝐹 “ 𝑅) = 𝑃)
8	7	reseq2d 5891	. . . 4 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 → (^◡𝐹 ↾ (𝐹 “ 𝑅)) = (^◡𝐹 ↾ 𝑃))
9	3, 8	sylan9eq 2798	. . 3 ⊢ ((Fun ^◡𝐹 ∧ (𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃) → ^◡(𝐹 ↾ 𝑅) = (^◡𝐹 ↾ 𝑃))
10	9	f1oeq1d 6711	. 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 231	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 396 = wceq 1539 ^◡ccnv 5588 ran crn 5590 ↾ cres 5591 “ cima 5592 Fun wfun 6427 –1-1→wf1 6430 –1-1-onto→wf1o 6432

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440

This theorem is referenced by: f1oresrab 6999 relogf1o 25722