Theorem rninxp 6183

Description: Two ways to express surjectivity of a restricted and corestricted binary relation (intersection of a binary relation with a Cartesian product). (Contributed by NM, 17-Jan-2006.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
rninxp	⊢ (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐶𝑦)

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

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem rninxp

Step	Hyp	Ref	Expression
1		dfss3 3902	. 2 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) ↔ ∀𝑦 ∈ 𝐵 𝑦 ∈ ran (𝐶 ↾ 𝐴))
2		ssrnres 6182	. 2 ⊢ (𝐵 ⊆ ran (𝐶 ↾ 𝐴) ↔ ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵)
3		df-ima 5668	. . . . 5 ⊢ (𝐶 “ 𝐴) = ran (𝐶 ↾ 𝐴)
4	3	eleq2i 2856	. . . 4 ⊢ (𝑦 ∈ (𝐶 “ 𝐴) ↔ 𝑦 ∈ ran (𝐶 ↾ 𝐴))
5		vex 3458	. . . . 5 ⊢ 𝑦 ∈ V
6	5	elima 6073	. . . 4 ⊢ (𝑦 ∈ (𝐶 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑥𝐶𝑦)
7	4, 6	bitr3i 280	. . 3 ⊢ (𝑦 ∈ ran (𝐶 ↾ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑥𝐶𝑦)
8	7	ralbii 3112	. 2 ⊢ (∀𝑦 ∈ 𝐵 𝑦 ∈ ran (𝐶 ↾ 𝐴) ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐶𝑦)
9	1, 2, 8	3bitr3i 304	1 ⊢ (ran (𝐶 ∩ (𝐴 × 𝐵)) = 𝐵 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑥𝐶𝑦)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   ∩ cin 4092   class class class wbr 5106   × cxp 5653  ran crn 5656   ↾ cres 5657   “ cima 5658

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-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-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  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-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668

This theorem is used by:  dminxp  6184  fncnv  6635  exfo  7129  brdom3  10634  brdom5  10635  brdom4  10636