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Theorem ssclem 16685

Description: Lemma for ssc1 16687 and similar theorems. (Contributed by Mario Carneiro, 6-Jan-2017.)

**Hypothesis**
Ref	Expression
isssc.1	⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))

**Assertion**
Ref	Expression
ssclem	⊢ (𝜑 → (𝐻 ∈ V ↔ 𝑆 ∈ V))

**Proof of Theorem ssclem**
Step	Hyp	Ref	Expression
1		dmxpid 5483	. . 3 ⊢ dom (𝑆 × 𝑆) = 𝑆
2		isssc.1	. . . . . . 7 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))
3		fndm 6130	. . . . . . 7 ⊢ (𝐻 Fn (𝑆 × 𝑆) → dom 𝐻 = (𝑆 × 𝑆))
4	2, 3	syl 17	. . . . . 6 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
5	4	adantr 466	. . . . 5 ⊢ ((𝜑 ∧ 𝐻 ∈ V) → dom 𝐻 = (𝑆 × 𝑆))
6		dmexg 7243	. . . . . 6 ⊢ (𝐻 ∈ V → dom 𝐻 ∈ V)
7	6	adantl 467	. . . . 5 ⊢ ((𝜑 ∧ 𝐻 ∈ V) → dom 𝐻 ∈ V)
8	5, 7	eqeltrrd 2850	. . . 4 ⊢ ((𝜑 ∧ 𝐻 ∈ V) → (𝑆 × 𝑆) ∈ V)
9		dmexg 7243	. . . 4 ⊢ ((𝑆 × 𝑆) ∈ V → dom (𝑆 × 𝑆) ∈ V)
10	8, 9	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝐻 ∈ V) → dom (𝑆 × 𝑆) ∈ V)
11	1, 10	syl5eqelr 2854	. 2 ⊢ ((𝜑 ∧ 𝐻 ∈ V) → 𝑆 ∈ V)
12		sqxpexg 7109	. . 3 ⊢ (𝑆 ∈ V → (𝑆 × 𝑆) ∈ V)
13		fnex 6624	. . 3 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ (𝑆 × 𝑆) ∈ V) → 𝐻 ∈ V)
14	2, 12, 13	syl2an 575	. 2 ⊢ ((𝜑 ∧ 𝑆 ∈ V) → 𝐻 ∈ V)
15	11, 14	impbida 794	1 ⊢ (𝜑 → (𝐻 ∈ V ↔ 𝑆 ∈ V))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1630 ∈ wcel 2144 Vcvv 3349 × cxp 5247 dom cdm 5249 Fn wfn 6026

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-rep 4902 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095

This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-ral 3065 df-rex 3066 df-reu 3067 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-op 4321 df-uni 4573 df-iun 4654 df-br 4785 df-opab 4845 df-mpt 4862 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039

This theorem is referenced by: ssc1 16687