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Mirrors > Home > MPE Home > Th. List > xpexgALT | Structured version Visualization version GIF version |
Description: Alternate proof of xpexg 7467 requiring Replacement (ax-rep 5182) but not Power Set (ax-pow 5258). (Contributed by Mario Carneiro, 20-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
xpexgALT | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunid 4976 | . . . 4 ⊢ ∪ 𝑦 ∈ 𝐵 {𝑦} = 𝐵 | |
2 | 1 | xpeq2i 5576 | . . 3 ⊢ (𝐴 × ∪ 𝑦 ∈ 𝐵 {𝑦}) = (𝐴 × 𝐵) |
3 | xpiundi 5616 | . . 3 ⊢ (𝐴 × ∪ 𝑦 ∈ 𝐵 {𝑦}) = ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) | |
4 | 2, 3 | eqtr3i 2846 | . 2 ⊢ (𝐴 × 𝐵) = ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) |
5 | id 22 | . . 3 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ 𝑊) | |
6 | fconstmpt 5608 | . . . . 5 ⊢ (𝐴 × {𝑦}) = (𝑥 ∈ 𝐴 ↦ 𝑦) | |
7 | mptexg 6978 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝑦) ∈ V) | |
8 | 6, 7 | eqeltrid 2917 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 × {𝑦}) ∈ V) |
9 | 8 | ralrimivw 3183 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∀𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V) |
10 | iunexg 7658 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ ∀𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V) → ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V) | |
11 | 5, 9, 10 | syl2anr 598 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ∈ V) |
12 | 4, 11 | eqeltrid 2917 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 {csn 4560 ∪ ciun 4911 ↦ cmpt 5138 × cxp 5547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 |
This theorem is referenced by: (None) |
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