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Theorem f1orescnv 6632

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 6629	. . 3 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 → ^◡(𝐹 ↾ 𝑅):𝑃–1-1-onto→𝑅)
2	1	adantl 484	. 2 ⊢ ((Fun ^◡𝐹 ∧ (𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃) → ^◡(𝐹 ↾ 𝑅):𝑃–1-1-onto→𝑅)
3		funcnvres 6434	. . . 4 ⊢ (Fun ^◡𝐹 → ^◡(𝐹 ↾ 𝑅) = (^◡𝐹 ↾ (𝐹 “ 𝑅)))
4		df-ima 5570	. . . . . 6 ⊢ (𝐹 “ 𝑅) = ran (𝐹 ↾ 𝑅)
5		dff1o5 6626	. . . . . . 7 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 ↔ ((𝐹 ↾ 𝑅):𝑅–1-1→𝑃 ∧ ran (𝐹 ↾ 𝑅) = 𝑃))
6	5	simprbi 499	. . . . . 6 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 → ran (𝐹 ↾ 𝑅) = 𝑃)
7	4, 6	syl5eq 2870	. . . . 5 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 → (𝐹 “ 𝑅) = 𝑃)
8	7	reseq2d 5855	. . . 4 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 → (^◡𝐹 ↾ (𝐹 “ 𝑅)) = (^◡𝐹 ↾ 𝑃))
9	3, 8	sylan9eq 2878	. . 3 ⊢ ((Fun ^◡𝐹 ∧ (𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃) → ^◡(𝐹 ↾ 𝑅) = (^◡𝐹 ↾ 𝑃))
10		f1oeq1 6606	. . 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 234	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 208 ∧ wa 398 = wceq 1537 ^◡ccnv 5556 ran crn 5558 ↾ cres 5559 “ cima 5560 Fun wfun 6351 –1-1→wf1 6354 –1-1-onto→wf1o 6356

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364

This theorem is referenced by: f1oresrab 6891 relogf1o 25152

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