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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 6319 | . . 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 1325 | . 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 1483 ∀wral 2912 ⊆ wss 3574 ↾ cres 5116 Fn wfn 5883 ‘cfv 5888 |
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-8 1992 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-pow 4843 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-sbc 3436 df-csb 3534 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-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-fv 5896 |
This theorem is referenced by: bnj1523 31139 |
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