f1orescnv - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > f1orescnv		Structured version Visualization version GIF version

Theorem f1orescnv 6605

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 6602	. . 3 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 → ^◡(𝐹 ↾ 𝑅):𝑃–1-1-onto→𝑅)
2	1	adantl 485	. 2 ⊢ ((Fun ^◡𝐹 ∧ (𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃) → ^◡(𝐹 ↾ 𝑅):𝑃–1-1-onto→𝑅)
3		funcnvres 6402	. . . 4 ⊢ (Fun ^◡𝐹 → ^◡(𝐹 ↾ 𝑅) = (^◡𝐹 ↾ (𝐹 “ 𝑅)))
4		df-ima 5532	. . . . . 6 ⊢ (𝐹 “ 𝑅) = ran (𝐹 ↾ 𝑅)
5		dff1o5 6599	. . . . . . 7 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 ↔ ((𝐹 ↾ 𝑅):𝑅–1-1→𝑃 ∧ ran (𝐹 ↾ 𝑅) = 𝑃))
6	5	simprbi 500	. . . . . 6 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 → ran (𝐹 ↾ 𝑅) = 𝑃)
7	4, 6	syl5eq 2845	. . . . 5 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 → (𝐹 “ 𝑅) = 𝑃)
8	7	reseq2d 5818	. . . 4 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 → (^◡𝐹 ↾ (𝐹 “ 𝑅)) = (^◡𝐹 ↾ 𝑃))
9	3, 8	sylan9eq 2853	. . 3 ⊢ ((Fun ^◡𝐹 ∧ (𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃) → ^◡(𝐹 ↾ 𝑅) = (^◡𝐹 ↾ 𝑃))
10		f1oeq1 6579	. . 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 235	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 209 ∧ wa 399 = wceq 1538 ^◡ccnv 5518 ran crn 5520 ↾ cres 5521 “ cima 5522 Fun wfun 6318 –1-1→wf1 6321 –1-1-onto→wf1o 6323

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331

This theorem is referenced by: f1oresrab 6866 relogf1o 25158

W3C validator