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Mirrors > Home > MPE Home > Th. List > ssclem | Structured version Visualization version GIF version |
Description: Lemma for ssc1 16908 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 5674 | . . 3 ⊢ dom (𝑆 × 𝑆) = 𝑆 | |
2 | isssc.1 | . . . . . . 7 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆)) | |
3 | fndm 6317 | . . . . . . 7 ⊢ (𝐻 Fn (𝑆 × 𝑆) → dom 𝐻 = (𝑆 × 𝑆)) | |
4 | 2, 3 | syl 17 | . . . . . 6 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆)) |
5 | 4 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐻 ∈ V) → dom 𝐻 = (𝑆 × 𝑆)) |
6 | dmexg 7460 | . . . . . 6 ⊢ (𝐻 ∈ V → dom 𝐻 ∈ V) | |
7 | 6 | adantl 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝐻 ∈ V) → dom 𝐻 ∈ V) |
8 | 5, 7 | eqeltrrd 2882 | . . . 4 ⊢ ((𝜑 ∧ 𝐻 ∈ V) → (𝑆 × 𝑆) ∈ V) |
9 | 8 | dmexd 7462 | . . 3 ⊢ ((𝜑 ∧ 𝐻 ∈ V) → dom (𝑆 × 𝑆) ∈ V) |
10 | 1, 9 | syl5eqelr 2886 | . 2 ⊢ ((𝜑 ∧ 𝐻 ∈ V) → 𝑆 ∈ V) |
11 | sqxpexg 7325 | . . 3 ⊢ (𝑆 ∈ V → (𝑆 × 𝑆) ∈ V) | |
12 | fnex 6837 | . . 3 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ (𝑆 × 𝑆) ∈ V) → 𝐻 ∈ V) | |
13 | 2, 11, 12 | syl2an 595 | . 2 ⊢ ((𝜑 ∧ 𝑆 ∈ V) → 𝐻 ∈ V) |
14 | 10, 13 | impbida 797 | 1 ⊢ (𝜑 → (𝐻 ∈ V ↔ 𝑆 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1520 ∈ wcel 2079 Vcvv 3432 × cxp 5433 dom cdm 5435 Fn wfn 6212 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-ral 3108 df-rex 3109 df-reu 3110 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-op 4473 df-uni 4740 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-id 5340 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 |
This theorem is referenced by: ssc1 16908 |
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