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Theorem bj-projeq 35778
Description: Substitution property for Proj. (Contributed by BJ, 6-Apr-2019.)
Assertion
Ref Expression
bj-projeq (𝐴 = 𝐶 → (𝐵 = 𝐷 → (𝐴 Proj 𝐵) = (𝐶 Proj 𝐷)))

Proof of Theorem bj-projeq
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simpr 486 . . . . . 6 ((𝐴 = 𝐶𝐵 = 𝐷) → 𝐵 = 𝐷)
2 simpl 484 . . . . . . 7 ((𝐴 = 𝐶𝐵 = 𝐷) → 𝐴 = 𝐶)
32sneqd 4636 . . . . . 6 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴} = {𝐶})
41, 3imaeq12d 6053 . . . . 5 ((𝐴 = 𝐶𝐵 = 𝐷) → (𝐵 “ {𝐴}) = (𝐷 “ {𝐶}))
54eleq2d 2820 . . . 4 ((𝐴 = 𝐶𝐵 = 𝐷) → ({𝑥} ∈ (𝐵 “ {𝐴}) ↔ {𝑥} ∈ (𝐷 “ {𝐶})))
65abbidv 2802 . . 3 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})} = {𝑥 ∣ {𝑥} ∈ (𝐷 “ {𝐶})})
7 df-bj-proj 35777 . . 3 (𝐴 Proj 𝐵) = {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})}
8 df-bj-proj 35777 . . 3 (𝐶 Proj 𝐷) = {𝑥 ∣ {𝑥} ∈ (𝐷 “ {𝐶})}
96, 7, 83eqtr4g 2798 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → (𝐴 Proj 𝐵) = (𝐶 Proj 𝐷))
109ex 414 1 (𝐴 = 𝐶 → (𝐵 = 𝐷 → (𝐴 Proj 𝐵) = (𝐶 Proj 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {cab 2710  {csn 4624  cima 5675   Proj bj-cproj 35776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5145  df-opab 5207  df-xp 5678  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-bj-proj 35777
This theorem is referenced by:  bj-projeq2  35779
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