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| Description: Lemma for marypha2 9480. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.) | 
| Ref | Expression | 
|---|---|
| marypha2lem.t | ⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) | 
| Ref | Expression | 
|---|---|
| marypha2lem1 | ⊢ 𝑇 ⊆ (𝐴 × ∪ ran 𝐹) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | marypha2lem.t | . 2 ⊢ 𝑇 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) | |
| 2 | iunss 5044 | . . 3 ⊢ (∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ⊆ (𝐴 × ∪ ran 𝐹) ↔ ∀𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ⊆ (𝐴 × ∪ ran 𝐹)) | |
| 3 | snssi 4807 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴) | |
| 4 | fvssunirn 6938 | . . . 4 ⊢ (𝐹‘𝑥) ⊆ ∪ ran 𝐹 | |
| 5 | xpss12 5699 | . . . 4 ⊢ (({𝑥} ⊆ 𝐴 ∧ (𝐹‘𝑥) ⊆ ∪ ran 𝐹) → ({𝑥} × (𝐹‘𝑥)) ⊆ (𝐴 × ∪ ran 𝐹)) | |
| 6 | 3, 4, 5 | sylancl 586 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ({𝑥} × (𝐹‘𝑥)) ⊆ (𝐴 × ∪ ran 𝐹)) | 
| 7 | 2, 6 | mprgbir 3067 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ⊆ (𝐴 × ∪ ran 𝐹) | 
| 8 | 1, 7 | eqsstri 4029 | 1 ⊢ 𝑇 ⊆ (𝐴 × ∪ ran 𝐹) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 {csn 4625 ∪ cuni 4906 ∪ ciun 4990 × cxp 5682 ran crn 5685 ‘cfv 6560 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-xp 5690 df-cnv 5692 df-dm 5694 df-rn 5695 df-iota 6513 df-fv 6568 | 
| This theorem is referenced by: marypha2 9480 | 
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