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Mirrors > Home > MPE Home > Th. List > marypha2lem1 | Structured version Visualization version GIF version |
Description: Lemma for marypha2 9383. 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 5009 | . . 3 ⊢ (∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ⊆ (𝐴 × ∪ ran 𝐹) ↔ ∀𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ⊆ (𝐴 × ∪ ran 𝐹)) | |
3 | snssi 4772 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → {𝑥} ⊆ 𝐴) | |
4 | fvssunirn 6879 | . . . 4 ⊢ (𝐹‘𝑥) ⊆ ∪ ran 𝐹 | |
5 | xpss12 5652 | . . . 4 ⊢ (({𝑥} ⊆ 𝐴 ∧ (𝐹‘𝑥) ⊆ ∪ ran 𝐹) → ({𝑥} × (𝐹‘𝑥)) ⊆ (𝐴 × ∪ ran 𝐹)) | |
6 | 3, 4, 5 | sylancl 587 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ({𝑥} × (𝐹‘𝑥)) ⊆ (𝐴 × ∪ ran 𝐹)) |
7 | 2, 6 | mprgbir 3068 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (𝐹‘𝑥)) ⊆ (𝐴 × ∪ ran 𝐹) |
8 | 1, 7 | eqsstri 3982 | 1 ⊢ 𝑇 ⊆ (𝐴 × ∪ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 ⊆ wss 3914 {csn 4590 ∪ cuni 4869 ∪ ciun 4958 × cxp 5635 ran crn 5638 ‘cfv 6500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-xp 5643 df-cnv 5645 df-dm 5647 df-rn 5648 df-iota 6452 df-fv 6508 |
This theorem is referenced by: marypha2 9383 |
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