Theorem disjimeceqbi2 39311

Description: Injectivity of the block constructor under disjointness. suc11reg 9576 analogue: under disjointness, equal blocks force equal generators (on dom 𝑅). (Contributed by Peter Mazsa, 16-Feb-2026.)

Assertion

Ref	Expression
disjimeceqbi2	⊢ ( Disj 𝑅 → ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → ([𝐴]𝑅 = [𝐵]𝑅 ↔ 𝐴 = 𝐵)))

Proof of Theorem disjimeceqbi2

Step	Hyp	Ref	Expression
1		disjimeceqim2 39309	. 2 ⊢ ( Disj 𝑅 → ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵)))
2		eceq1 8720	. . 3 ⊢ (𝐴 = 𝐵 → [𝐴]𝑅 = [𝐵]𝑅)
3	2	2a1i 12	. 2 ⊢ ( Disj 𝑅 → ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → (𝐴 = 𝐵 → [𝐴]𝑅 = [𝐵]𝑅)))
4	1, 3	impbidd 212	1 ⊢ ( Disj 𝑅 → ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → ([𝐴]𝑅 = [𝐵]𝑅 ↔ 𝐴 = 𝐵)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562   ∈ wcel 2144  dom cdm 5649  [cec 8678   Disj wdisjALTV 38723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5103  df-opab 5165  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-ec 8682  df-coss 39005  df-cnvrefrel 39111  df-disjALTV 39294

This theorem is referenced by: (None)