Theorem disjrnmpt 32660

Description: Rewriting a disjoint collection using the range of a mapping. (Contributed by Thierry Arnoux, 27-May-2020.)

Assertion

Ref	Expression
disjrnmpt	⊢ (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦)

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

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem disjrnmpt

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		disjabrex 32657	. 2 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}𝑦)
2		eqid 2736	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	2	rnmpt 5906	. . 3 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}
4		disjeq1 5072	. . 3 ⊢ (ran (𝑥 ∈ 𝐴 ↦ 𝐵) = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} → (Disj 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ↔ Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}𝑦))
5	3, 4	ax-mp 5	. 2 ⊢ (Disj 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦 ↔ Disj 𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}𝑦)
6	1, 5	sylibr 234	1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ 𝐵)𝑦)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541  {cab 2714  ∃wrex 3060  Disj wdisj 5065   ↦ cmpt 5179  ran crn 5625

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 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-disj 5066  df-br 5099  df-opab 5161  df-mpt 5180  df-cnv 5632  df-dm 5634  df-rn 5635

This theorem is referenced by:  fnpreimac  32749  sigapildsys  34319  ldgenpisyslem1  34320  carsgclctunlem2  34476  pmeasadd  34482