Theorem disjimeceqbi 39930

Description: Disj gives biconditional injectivity (domain-wise). Strengthens injectivity to an iff. (Contributed by Peter Mazsa, 3-Feb-2026.)

Assertion

Ref	Expression
disjimeceqbi	⊢ ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 ↔ 𝑢 = 𝑣))

Distinct variable group:   𝑢,𝑅,𝑣

Proof of Theorem disjimeceqbi

Step	Hyp	Ref	Expression
1		disjimeceqim 39928	. 2 ⊢ ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣))
2		eceq1 8785	. . 3 ⊢ (𝑢 = 𝑣 → [𝑢]𝑅 = [𝑣]𝑅)
3	2	rgen2w 3085	. 2 ⊢ ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 → [𝑢]𝑅 = [𝑣]𝑅)
4		2ralbiim 3145	. 2 ⊢ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 ↔ 𝑢 = 𝑣) ↔ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣) ∧ ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 → [𝑢]𝑅 = [𝑣]𝑅)))
5	1, 3, 4	sylanblrc 602	1 ⊢ ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 ↔ 𝑢 = 𝑣))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  ∀wral 3080  dom cdm 5655  [cec 8743   Disj wdisjALTV 39343

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ec 8747  df-coss 39625  df-cnvrefrel 39731  df-disjALTV 39914

This theorem is used by: (None)