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Mirrors > Home > MPE Home > Th. List > ssclem | Structured version Visualization version GIF version |
Description: Lemma for ssc1 17869 and similar theorems. (Contributed by Mario Carneiro, 6-Jan-2017.) |
Ref | Expression |
---|---|
isssc.1 | ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆)) |
Ref | Expression |
---|---|
ssclem | ⊢ (𝜑 → (𝐻 ∈ V ↔ 𝑆 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmxpid 5944 | . . 3 ⊢ dom (𝑆 × 𝑆) = 𝑆 | |
2 | isssc.1 | . . . . . . 7 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆)) | |
3 | 2 | fndmd 6674 | . . . . . 6 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆)) |
4 | 3 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐻 ∈ V) → dom 𝐻 = (𝑆 × 𝑆)) |
5 | dmexg 7924 | . . . . . 6 ⊢ (𝐻 ∈ V → dom 𝐻 ∈ V) | |
6 | 5 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐻 ∈ V) → dom 𝐻 ∈ V) |
7 | 4, 6 | eqeltrrd 2840 | . . . 4 ⊢ ((𝜑 ∧ 𝐻 ∈ V) → (𝑆 × 𝑆) ∈ V) |
8 | 7 | dmexd 7926 | . . 3 ⊢ ((𝜑 ∧ 𝐻 ∈ V) → dom (𝑆 × 𝑆) ∈ V) |
9 | 1, 8 | eqeltrrid 2844 | . 2 ⊢ ((𝜑 ∧ 𝐻 ∈ V) → 𝑆 ∈ V) |
10 | sqxpexg 7774 | . . 3 ⊢ (𝑆 ∈ V → (𝑆 × 𝑆) ∈ V) | |
11 | fnex 7237 | . . 3 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ (𝑆 × 𝑆) ∈ V) → 𝐻 ∈ V) | |
12 | 2, 10, 11 | syl2an 596 | . 2 ⊢ ((𝜑 ∧ 𝑆 ∈ V) → 𝐻 ∈ V) |
13 | 9, 12 | impbida 801 | 1 ⊢ (𝜑 → (𝐻 ∈ V ↔ 𝑆 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 × cxp 5687 dom cdm 5689 Fn wfn 6558 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 |
This theorem is referenced by: ssc1 17869 |
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