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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 5931 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ⊆ dom 𝐺) → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴)) | |
5 | 1, 2, 3, 4 | syl3anc 1326 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ⊆ wss 3574 dom cdm 5114 ↾ cres 5116 Fun wfun 5882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-res 5126 df-fun 5890 |
This theorem is referenced by: bnj1442 31117 bnj1450 31118 bnj1501 31135 |
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