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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1503 | Structured version Visualization version GIF version |
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
Ref | Expression |
---|---|
bnj1503.1 | ⊢ (𝜑 → Fun 𝐹) |
bnj1503.2 | ⊢ (𝜑 → 𝐺 ⊆ 𝐹) |
bnj1503.3 | ⊢ (𝜑 → 𝐴 ⊆ dom 𝐺) |
Ref | Expression |
---|---|
bnj1503 | ⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1503.1 | . 2 ⊢ (𝜑 → Fun 𝐹) | |
2 | bnj1503.2 | . 2 ⊢ (𝜑 → 𝐺 ⊆ 𝐹) | |
3 | bnj1503.3 | . 2 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐺) | |
4 | fun2ssres 6473 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ⊆ dom 𝐺) → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴)) | |
5 | 1, 2, 3, 4 | syl3anc 1370 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ⊆ wss 3888 dom cdm 5586 ↾ cres 5588 Fun wfun 6422 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3433 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-if 4462 df-sn 4564 df-pr 4566 df-op 4570 df-br 5076 df-opab 5138 df-id 5486 df-xp 5592 df-rel 5593 df-cnv 5594 df-co 5595 df-dm 5596 df-res 5598 df-fun 6430 |
This theorem is referenced by: bnj1442 33016 bnj1450 33017 bnj1501 33034 |
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