Theorem f1orescnv 6849

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 6846	. . 3 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 → ◡(𝐹 ↾ 𝑅):𝑃–1-1-onto→𝑅)
2	1	adantl 480	. 2 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃) → ◡(𝐹 ↾ 𝑅):𝑃–1-1-onto→𝑅)
3		funcnvres 6626	. . . 4 ⊢ (Fun ◡𝐹 → ◡(𝐹 ↾ 𝑅) = (◡𝐹 ↾ (𝐹 “ 𝑅)))
4		df-ima 5685	. . . . . 6 ⊢ (𝐹 “ 𝑅) = ran (𝐹 ↾ 𝑅)
5		dff1o5 6843	. . . . . . 7 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 ↔ ((𝐹 ↾ 𝑅):𝑅–1-1→𝑃 ∧ ran (𝐹 ↾ 𝑅) = 𝑃))
6	5	simprbi 495	. . . . . 6 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 → ran (𝐹 ↾ 𝑅) = 𝑃)
7	4, 6	eqtrid 2777	. . . . 5 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 → (𝐹 “ 𝑅) = 𝑃)
8	7	reseq2d 5979	. . . 4 ⊢ ((𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃 → (◡𝐹 ↾ (𝐹 “ 𝑅)) = (◡𝐹 ↾ 𝑃))
9	3, 8	sylan9eq 2785	. . 3 ⊢ ((Fun ◡𝐹 ∧ (𝐹 ↾ 𝑅):𝑅–1-1-onto→𝑃) → ◡(𝐹 ↾ 𝑅) = (◡𝐹 ↾ 𝑃))
10	9	f1oeq1d 6829	. 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)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533  ^◡ccnv 5671  ran crn 5673   ↾ cres 5674   “ cima 5675  Fun wfun 6537  –1-1→wf1 6540  –1-1-onto→wf1o 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550

This theorem is referenced by:  f1oresrab  7132  relogf1o  26518