Theorem mpoeq12 7428

Description: An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)

Assertion

Ref	Expression
mpoeq12	⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝐸))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦

Allowed substitution hints:   𝐸(𝑥,𝑦)

Proof of Theorem mpoeq12

Step	Hyp	Ref	Expression
1		eqid 2733	. . . . 5 ⊢ 𝐸 = 𝐸
2	1	rgenw 3053	. . . 4 ⊢ ∀𝑦 ∈ 𝐵 𝐸 = 𝐸
3	2	jctr 524	. . 3 ⊢ (𝐵 = 𝐷 → (𝐵 = 𝐷 ∧ ∀𝑦 ∈ 𝐵 𝐸 = 𝐸))
4	3	ralrimivw 3130	. 2 ⊢ (𝐵 = 𝐷 → ∀𝑥 ∈ 𝐴 (𝐵 = 𝐷 ∧ ∀𝑦 ∈ 𝐵 𝐸 = 𝐸))
5		mpoeq123 7427	. 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 593	1 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝐸))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  ∀wral 3049   ∈ cmpo 7357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ral 3050  df-oprab 7359  df-mpo 7360

This theorem is referenced by:  dffi3  9325  cantnfres  9577  xpsval  17484  monpropd  17654  grpsubpropd2  18969  lsmvalx  19561  lsmpropd  19599  funcrngcsetc  20565  funcringcsetc  20599  psrplusgpropd  22158  d1mat2pmat  22664  txval  23489  cnmptk1p  23610  cnmptk2  23611  xpstopnlem1  23734  rrxmval  25342  madjusmdetlem1  33851  pstmval  33919  qqhval2  34006  lmod1zr  48608  infsubc2d  49177