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Theorem bj-projeq 33311
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 471 . . . . . 6 ((𝐴 = 𝐶𝐵 = 𝐷) → 𝐵 = 𝐷)
2 simpl 468 . . . . . . 7 ((𝐴 = 𝐶𝐵 = 𝐷) → 𝐴 = 𝐶)
32sneqd 4328 . . . . . 6 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴} = {𝐶})
41, 3imaeq12d 5608 . . . . 5 ((𝐴 = 𝐶𝐵 = 𝐷) → (𝐵 “ {𝐴}) = (𝐷 “ {𝐶}))
54eleq2d 2836 . . . 4 ((𝐴 = 𝐶𝐵 = 𝐷) → ({𝑥} ∈ (𝐵 “ {𝐴}) ↔ {𝑥} ∈ (𝐷 “ {𝐶})))
65abbidv 2890 . . 3 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})} = {𝑥 ∣ {𝑥} ∈ (𝐷 “ {𝐶})})
7 df-bj-proj 33310 . . 3 (𝐴 Proj 𝐵) = {𝑥 ∣ {𝑥} ∈ (𝐵 “ {𝐴})}
8 df-bj-proj 33310 . . 3 (𝐶 Proj 𝐷) = {𝑥 ∣ {𝑥} ∈ (𝐷 “ {𝐶})}
96, 7, 83eqtr4g 2830 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → (𝐴 Proj 𝐵) = (𝐶 Proj 𝐷))
109ex 397 1 (𝐴 = 𝐶 → (𝐵 = 𝐷 → (𝐴 Proj 𝐵) = (𝐶 Proj 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  {cab 2757  {csn 4316  cima 5252   Proj bj-cproj 33309
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-bj-proj 33310
This theorem is referenced by:  bj-projeq2  33312
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