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Theorem mapsnf1o3 8896

Description: Explicit bijection in the reverse of mapsnf1o2 8895. (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 2770	. . . 4 ⊢ (𝑥 ∈ (𝐵 ↑_m 𝑆) ↦ (𝑥‘𝑋)) = (𝑥 ∈ (𝐵 ↑_m 𝑆) ↦ (𝑥‘𝑋))
5	1, 2, 3, 4	mapsnf1o2 8895	. . 3 ⊢ (𝑥 ∈ (𝐵 ↑_m 𝑆) ↦ (𝑥‘𝑋)):(𝐵 ↑_m 𝑆)–1-1-onto→𝐵
6		f1ocnv 6837	. . 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 8894	. . . 4 ⊢ ^◡(𝑥 ∈ (𝐵 ↑_m 𝑆) ↦ (𝑥‘𝑋)) = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦}))
10	8, 9	eqtr4i 2796	. . 3 ⊢ 𝐹 = ^◡(𝑥 ∈ (𝐵 ↑_m 𝑆) ↦ (𝑥‘𝑋))
11		f1oeq1 6812	. . 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 234	1 ⊢ 𝐹:𝐵–1-1-onto→(𝐵 ↑_m 𝑆)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 209 = wceq 1568 ∈ wcel 2150 Vcvv 3462 {csn 4594 ↦ cmpt 5197 × cxp 5663 ^◡ccnv 5664 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7414 ↑_m cmap 8827

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2152 ax-9 2160 ax-10 2183 ax-11 2199 ax-12 2220 ax-ext 2742 ax-sep 5262 ax-nul 5274 ax-pow 5340 ax-pr 5408 ax-un 7736

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2099 df-mo 2574 df-eu 2604 df-clab 2749 df-cleq 2762 df-clel 2845 df-nfc 2919 df-ne 2966 df-ral 3087 df-rex 3097 df-reu 3377 df-rab 3424 df-v 3464 df-sbc 3753 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-iun 4963 df-br 5115 df-opab 5179 df-mpt 5198 df-id 5560 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7989 df-2nd 7990 df-map 8829

This theorem is referenced by: coe1f2 22352 coe1add 22408 evls1rhmlem 22464 evl1sca 22477 pf1ind 22498 ismrer1 38437

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