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Mathbox for Jonathan Ben-Naim |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1536 | 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 |
---|---|
bnj1536.1 | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
bnj1536.2 | ⊢ (𝜑 → 𝐺 Fn 𝐴) |
bnj1536.3 | ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
bnj1536.4 | ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)) |
Ref | Expression |
---|---|
bnj1536 | ⊢ (𝜑 → (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bnj1536.4 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)) | |
2 | bnj1536.1 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴) | |
3 | bnj1536.2 | . . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴) | |
4 | bnj1536.3 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
5 | fvreseq 6359 | . . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐵 ⊆ 𝐴) → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥))) | |
6 | 2, 3, 4, 5 | syl21anc 1365 | . 2 ⊢ (𝜑 → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥))) |
7 | 1, 6 | mpbird 247 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1523 ∀wral 2941 ⊆ wss 3607 ↾ cres 5145 Fn wfn 5921 ‘cfv 5926 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-fv 5934 |
This theorem is referenced by: bnj1523 31265 |
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