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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 6838 | . . 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 838 | . 2 ⊢ (𝜑 → ((𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥))) |
7 | 1, 6 | mpbird 260 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝐵) = (𝐺 ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∀wral 3051 ⊆ wss 3853 ↾ cres 5538 Fn wfn 6353 ‘cfv 6358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-fv 6366 |
This theorem is referenced by: bnj1523 32718 |
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