elxpcbasex1ALT - Mathbox for Zhi Wang

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Theorem elxpcbasex1ALT 48928

Description: Alternate proof of elxpcbasex1 48927. (Contributed by Zhi Wang, 8-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
elxpcbasex1.t	⊢ 𝑇 = (𝐶 ×_c 𝐷)
elxpcbasex1.b	⊢ 𝐵 = (Base‘𝑇)
elxpcbasex1.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

**Assertion**
Ref	Expression
elxpcbasex1ALT	⊢ (𝜑 → 𝐶 ∈ V)

**Proof of Theorem elxpcbasex1ALT**
Step	Hyp	Ref	Expression
1		elxpcbasex1.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
2		elxpcbasex1.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑇)
3		elxpcbasex1.t	. . . . . 6 ⊢ 𝑇 = (𝐶 ×_c 𝐷)
4		eqid 2736	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
5		eqid 2736	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
6	3, 4, 5	xpcbas 18219	. . . . 5 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑇)
7	2, 6	eqtr4i 2767	. . . 4 ⊢ 𝐵 = ((Base‘𝐶) × (Base‘𝐷))
8	1, 7	eleqtrdi 2850	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((Base‘𝐶) × (Base‘𝐷)))
9		xp1st 8042	. . 3 ⊢ (𝑋 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (1^st ‘𝑋) ∈ (Base‘𝐶))
10	8, 9	syl 17	. 2 ⊢ (𝜑 → (1^st ‘𝑋) ∈ (Base‘𝐶))
11	10	elfvexd 6943	1 ⊢ (𝜑 → 𝐶 ∈ V)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3479 × cxp 5681 ‘cfv 6559 (class class class)co 7429 1^st c1st 8008 Basecbs 17243 ×_c cxpc 18209

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-om 7884 df-1st 8010 df-2nd 8011 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-er 8741 df-en 8982 df-dom 8983 df-sdom 8984 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-nn 12263 df-2 12325 df-3 12326 df-4 12327 df-5 12328 df-6 12329 df-7 12330 df-8 12331 df-9 12332 df-n0 12523 df-z 12610 df-dec 12730 df-slot 17215 df-ndx 17227 df-base 17244 df-hom 17317 df-cco 17318 df-xpc 18213

This theorem is referenced by: (None)

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