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Mirrors > Home > MPE Home > Th. List > ssclem | Structured version Visualization version GIF version |
Description: Lemma for ssc1 17812 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 5932 | . . 3 ⊢ dom (𝑆 × 𝑆) = 𝑆 | |
2 | isssc.1 | . . . . . . 7 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆)) | |
3 | 2 | fndmd 6660 | . . . . . 6 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆)) |
4 | 3 | adantr 479 | . . . . 5 ⊢ ((𝜑 ∧ 𝐻 ∈ V) → dom 𝐻 = (𝑆 × 𝑆)) |
5 | dmexg 7909 | . . . . . 6 ⊢ (𝐻 ∈ V → dom 𝐻 ∈ V) | |
6 | 5 | adantl 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐻 ∈ V) → dom 𝐻 ∈ V) |
7 | 4, 6 | eqeltrrd 2826 | . . . 4 ⊢ ((𝜑 ∧ 𝐻 ∈ V) → (𝑆 × 𝑆) ∈ V) |
8 | 7 | dmexd 7911 | . . 3 ⊢ ((𝜑 ∧ 𝐻 ∈ V) → dom (𝑆 × 𝑆) ∈ V) |
9 | 1, 8 | eqeltrrid 2830 | . 2 ⊢ ((𝜑 ∧ 𝐻 ∈ V) → 𝑆 ∈ V) |
10 | sqxpexg 7758 | . . 3 ⊢ (𝑆 ∈ V → (𝑆 × 𝑆) ∈ V) | |
11 | fnex 7229 | . . 3 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ (𝑆 × 𝑆) ∈ V) → 𝐻 ∈ V) | |
12 | 2, 10, 11 | syl2an 594 | . 2 ⊢ ((𝜑 ∧ 𝑆 ∈ V) → 𝐻 ∈ V) |
13 | 9, 12 | impbida 799 | 1 ⊢ (𝜑 → (𝐻 ∈ V ↔ 𝑆 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 Vcvv 3461 × cxp 5676 dom cdm 5678 Fn wfn 6544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 |
This theorem is referenced by: ssc1 17812 |
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