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Mirrors > Home > NFE Home > Th. List > mpt2eq12 | GIF version |
Description: An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) |
Ref | Expression |
---|---|
mpt2eq12 | ⊢ ((A = C ∧ B = D) → (x ∈ A, y ∈ B ↦ E) = (x ∈ C, y ∈ D ↦ E)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2353 | . . . . 5 ⊢ E = E | |
2 | 1 | rgenw 2681 | . . . 4 ⊢ ∀y ∈ B E = E |
3 | 2 | jctr 526 | . . 3 ⊢ (B = D → (B = D ∧ ∀y ∈ B E = E)) |
4 | 3 | ralrimivw 2698 | . 2 ⊢ (B = D → ∀x ∈ A (B = D ∧ ∀y ∈ B E = E)) |
5 | mpt2eq123 5661 | . 2 ⊢ ((A = C ∧ ∀x ∈ A (B = D ∧ ∀y ∈ B E = E)) → (x ∈ A, y ∈ B ↦ E) = (x ∈ C, y ∈ D ↦ E)) | |
6 | 4, 5 | sylan2 460 | 1 ⊢ ((A = C ∧ B = D) → (x ∈ A, y ∈ B ↦ E) = (x ∈ C, y ∈ D ↦ E)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 = wceq 1642 ∀wral 2614 ↦ cmpt2 5653 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ral 2619 df-oprab 5528 df-mpt2 5654 |
This theorem is referenced by: (None) |
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