|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > ssclem | Structured version Visualization version GIF version | ||
| Description: Lemma for ssc1 17865 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 5941 | . . 3 ⊢ dom (𝑆 × 𝑆) = 𝑆 | |
| 2 | isssc.1 | . . . . . . 7 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆)) | |
| 3 | 2 | fndmd 6673 | . . . . . 6 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆)) | 
| 4 | 3 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝐻 ∈ V) → dom 𝐻 = (𝑆 × 𝑆)) | 
| 5 | dmexg 7923 | . . . . . 6 ⊢ (𝐻 ∈ V → dom 𝐻 ∈ V) | |
| 6 | 5 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐻 ∈ V) → dom 𝐻 ∈ V) | 
| 7 | 4, 6 | eqeltrrd 2842 | . . . 4 ⊢ ((𝜑 ∧ 𝐻 ∈ V) → (𝑆 × 𝑆) ∈ V) | 
| 8 | 7 | dmexd 7925 | . . 3 ⊢ ((𝜑 ∧ 𝐻 ∈ V) → dom (𝑆 × 𝑆) ∈ V) | 
| 9 | 1, 8 | eqeltrrid 2846 | . 2 ⊢ ((𝜑 ∧ 𝐻 ∈ V) → 𝑆 ∈ V) | 
| 10 | sqxpexg 7775 | . . 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 1540 ∈ wcel 2108 Vcvv 3480 × cxp 5683 dom cdm 5685 Fn wfn 6556 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 | 
| This theorem is referenced by: ssc1 17865 | 
| Copyright terms: Public domain | W3C validator |