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Theorem mapsnf1o3 8888

Description: Explicit bijection in the reverse of mapsnf1o2 8887. (Contributed by Stefan O'Rear, 24-Mar-2015.)

**Hypotheses**
Ref	Expression
mapsncnv.s	⊢ 𝑆 = {𝑋}
mapsncnv.b	⊢ 𝐵 ∈ V
mapsncnv.x	⊢ 𝑋 ∈ V
mapsnf1o3.f	⊢ 𝐹 = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦}))

**Assertion**
Ref	Expression
mapsnf1o3	⊢ 𝐹:𝐵–1-1-onto→(𝐵 ↑_m 𝑆)

Distinct variable groups: 𝑦,𝐵 𝑦,𝑆 𝑦,𝑋 Allowed substitution hint: 𝐹(𝑦)

Proof of Theorem mapsnf1o3 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mapsncnv.s	. . . 4 ⊢ 𝑆 = {𝑋}
2		mapsncnv.b	. . . 4 ⊢ 𝐵 ∈ V
3		mapsncnv.x	. . . 4 ⊢ 𝑋 ∈ V
4		eqid 2732	. . . 4 ⊢ (𝑥 ∈ (𝐵 ↑_m 𝑆) ↦ (𝑥‘𝑋)) = (𝑥 ∈ (𝐵 ↑_m 𝑆) ↦ (𝑥‘𝑋))
5	1, 2, 3, 4	mapsnf1o2 8887	. . 3 ⊢ (𝑥 ∈ (𝐵 ↑_m 𝑆) ↦ (𝑥‘𝑋)):(𝐵 ↑_m 𝑆)–1-1-onto→𝐵
6		f1ocnv 6845	. . 3 ⊢ ((𝑥 ∈ (𝐵 ↑_m 𝑆) ↦ (𝑥‘𝑋)):(𝐵 ↑_m 𝑆)–1-1-onto→𝐵 → ^◡(𝑥 ∈ (𝐵 ↑_m 𝑆) ↦ (𝑥‘𝑋)):𝐵–1-1-onto→(𝐵 ↑_m 𝑆))
7	5, 6	ax-mp 5	. 2 ⊢ ^◡(𝑥 ∈ (𝐵 ↑_m 𝑆) ↦ (𝑥‘𝑋)):𝐵–1-1-onto→(𝐵 ↑_m 𝑆)
8		mapsnf1o3.f	. . . 4 ⊢ 𝐹 = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦}))
9	1, 2, 3, 4	mapsncnv 8886	. . . 4 ⊢ ^◡(𝑥 ∈ (𝐵 ↑_m 𝑆) ↦ (𝑥‘𝑋)) = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦}))
10	8, 9	eqtr4i 2763	. . 3 ⊢ 𝐹 = ^◡(𝑥 ∈ (𝐵 ↑_m 𝑆) ↦ (𝑥‘𝑋))
11		f1oeq1 6821	. . 3 ⊢ (𝐹 = ^◡(𝑥 ∈ (𝐵 ↑_m 𝑆) ↦ (𝑥‘𝑋)) → (𝐹:𝐵–1-1-onto→(𝐵 ↑_m 𝑆) ↔ ^◡(𝑥 ∈ (𝐵 ↑_m 𝑆) ↦ (𝑥‘𝑋)):𝐵–1-1-onto→(𝐵 ↑_m 𝑆)))
12	10, 11	ax-mp 5	. 2 ⊢ (𝐹:𝐵–1-1-onto→(𝐵 ↑_m 𝑆) ↔ ^◡(𝑥 ∈ (𝐵 ↑_m 𝑆) ↦ (𝑥‘𝑋)):𝐵–1-1-onto→(𝐵 ↑_m 𝑆))
13	7, 12	mpbir 230	1 ⊢ 𝐹:𝐵–1-1-onto→(𝐵 ↑_m 𝑆)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 = wceq 1541 ∈ wcel 2106 Vcvv 3474 {csn 4628 ↦ cmpt 5231 × cxp 5674 ^◡ccnv 5675 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7408 ↑_m cmap 8819

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-map 8821

This theorem is referenced by: coe1f2 21732 coe1add 21785 evls1rhmlem 21839 evl1sca 21852 pf1ind 21873 ismrer1 36701

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