Theorem disjimeceqbi 39145

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 39143	. 2 ⊢ ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣))
2		eceq1 8678	. . 3 ⊢ (𝑢 = 𝑣 → [𝑢]𝑅 = [𝑣]𝑅)
3	2	rgen2w 3057	. 2 ⊢ ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(𝑢 = 𝑣 → [𝑢]𝑅 = [𝑣]𝑅)
4		2ralbiim 3117	. 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 591	1 ⊢ ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 ↔ 𝑢 = 𝑣))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542  ∀wral 3052  dom cdm 5626  [cec 8636   Disj wdisjALTV 38558

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 5232  ax-pr 5372

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 3063  df-rmo 3343  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-ec 8640  df-coss 38840  df-cnvrefrel 38946  df-disjALTV 39129

This theorem is referenced by: (None)