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Theorem bj-projeq 32173
 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 476 . . . . . 6 ((𝐴 = 𝐶𝐵 = 𝐷) → 𝐵 = 𝐷)
2 simpl 472 . . . . . . 7 ((𝐴 = 𝐶𝐵 = 𝐷) → 𝐴 = 𝐶)
32sneqd 4137 . . . . . 6 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴} = {𝐶})
41, 3imaeq12d 5386 . . . . 5 ((𝐴 = 𝐶𝐵 = 𝐷) → (𝐵 “ {𝐴}) = (𝐷 “ {𝐶}))
54eleq2d 2673 . . . 4 ((𝐴 = 𝐶𝐵 = 𝐷) → ({𝑥} ∈ (𝐵 “ {𝐴}) ↔ {𝑥} ∈ (𝐷 “ {𝐶})))
65abbidv 2728 . . 3 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})} = {𝑥 ∣ {𝑥} ∈ (𝐷 “ {𝐶})})
7 df-bj-proj 32172 . . 3 (𝐴 Proj 𝐵) = {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})}
8 df-bj-proj 32172 . . 3 (𝐶 Proj 𝐷) = {𝑥 ∣ {𝑥} ∈ (𝐷 “ {𝐶})}
96, 7, 83eqtr4g 2669 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → (𝐴 Proj 𝐵) = (𝐶 Proj 𝐷))
109ex 449 1 (𝐴 = 𝐶 → (𝐵 = 𝐷 → (𝐴 Proj 𝐵) = (𝐶 Proj 𝐷)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  {csn 4125   “ cima 5041   Proj bj-cproj 32171 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-bj-proj 32172 This theorem is referenced by:  bj-projeq2  32174
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