Theorem disjimeceqbi2 39010

Description: Injectivity of the block constructor under disjointness. suc11reg 9532 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 39008	. 2 ⊢ ( Disj 𝑅 → ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → ([𝐴]𝑅 = [𝐵]𝑅 → 𝐴 = 𝐵)))
2		eceq1 8677	. . 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 210	1 ⊢ ( Disj 𝑅 → ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → ([𝐴]𝑅 = [𝐵]𝑅 ↔ 𝐴 = 𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  dom cdm 5625  [cec 8635   Disj wdisjALTV 38422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ec 8639  df-coss 38704  df-cnvrefrel 38810  df-disjALTV 38993

This theorem is referenced by: (None)