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Theorem bj-projeq 36329
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 484 . . . . . 6 ((𝐴 = 𝐶𝐵 = 𝐷) → 𝐵 = 𝐷)
2 simpl 482 . . . . . . 7 ((𝐴 = 𝐶𝐵 = 𝐷) → 𝐴 = 𝐶)
32sneqd 4632 . . . . . 6 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴} = {𝐶})
41, 3imaeq12d 6050 . . . . 5 ((𝐴 = 𝐶𝐵 = 𝐷) → (𝐵 “ {𝐴}) = (𝐷 “ {𝐶}))
54eleq2d 2811 . . . 4 ((𝐴 = 𝐶𝐵 = 𝐷) → ({𝑥} ∈ (𝐵 “ {𝐴}) ↔ {𝑥} ∈ (𝐷 “ {𝐶})))
65abbidv 2793 . . 3 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})} = {𝑥 ∣ {𝑥} ∈ (𝐷 “ {𝐶})})
7 df-bj-proj 36328 . . 3 (𝐴 Proj 𝐵) = {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})}
8 df-bj-proj 36328 . . 3 (𝐶 Proj 𝐷) = {𝑥 ∣ {𝑥} ∈ (𝐷 “ {𝐶})}
96, 7, 83eqtr4g 2789 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → (𝐴 Proj 𝐵) = (𝐶 Proj 𝐷))
109ex 412 1 (𝐴 = 𝐶 → (𝐵 = 𝐷 → (𝐴 Proj 𝐵) = (𝐶 Proj 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  {cab 2701  {csn 4620  cima 5669   Proj bj-cproj 36327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-xp 5672  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-bj-proj 36328
This theorem is referenced by:  bj-projeq2  36330
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