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Theorem mapsnf1o3 8838

Description: Explicit bijection in the reverse of mapsnf1o2 8837. (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 2737	. . . 4 ⊢ (𝑥 ∈ (𝐵 ↑_m 𝑆) ↦ (𝑥‘𝑋)) = (𝑥 ∈ (𝐵 ↑_m 𝑆) ↦ (𝑥‘𝑋))
5	1, 2, 3, 4	mapsnf1o2 8837	. . 3 ⊢ (𝑥 ∈ (𝐵 ↑_m 𝑆) ↦ (𝑥‘𝑋)):(𝐵 ↑_m 𝑆)–1-1-onto→𝐵
6		f1ocnv 6787	. . 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 8836	. . . 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 6763	. . 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 231	1 ⊢ 𝐹:𝐵–1-1-onto→(𝐵 ↑_m 𝑆)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 = wceq 1542 ∈ wcel 2114 Vcvv 3441 {csn 4581 ↦ cmpt 5180 × cxp 5623 ^◡ccnv 5624 –1-1-onto→wf1o 6492 ‘cfv 6493 (class class class)co 7361 ↑_m cmap 8768

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7683

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7364 df-oprab 7365 df-mpo 7366 df-1st 7936 df-2nd 7937 df-map 8770

This theorem is referenced by: coe1f2 22155 coe1add 22211 evls1rhmlem 22270 evl1sca 22283 pf1ind 22304 ismrer1 38052

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