Theorem elrnmpt2 5537

Description: Membership in the range of an operation class abstraction. (Contributed by NM, 1-Aug-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypotheses

Ref	Expression
rngop.1	⊢ 𝐹 = (x ∈ A, y ∈ B ↦ 𝐶)
elrnmpt2.1	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
elrnmpt2	⊢ (𝐷 ∈ ran 𝐹 ↔ ∃x ∈ A ∃y ∈ B 𝐷 = 𝐶)

Distinct variable groups:   y,A   x,y,𝐷

Allowed substitution hints:   A(x)   B(x,y)   𝐶(x,y)   𝐹(x,y)

Proof of Theorem elrnmpt2

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rngop.1	. . . 4 ⊢ 𝐹 = (x ∈ A, y ∈ B ↦ 𝐶)
2	1	rnmpt2 5534	. . 3 ⊢ ran 𝐹 = {z ∣ ∃x ∈ A ∃y ∈ B z = 𝐶}
3	2	eleq2i 2086	. 2 ⊢ (𝐷 ∈ ran 𝐹 ↔ 𝐷 ∈ {z ∣ ∃x ∈ A ∃y ∈ B z = 𝐶})
4		elrnmpt2.1	. . . . . 6 ⊢ 𝐶 ∈ V
5		eleq1 2082	. . . . . 6 ⊢ (𝐷 = 𝐶 → (𝐷 ∈ V ↔ 𝐶 ∈ V))
6	4, 5	mpbiri 157	. . . . 5 ⊢ (𝐷 = 𝐶 → 𝐷 ∈ V)
7	6	rexlimivw 2407	. . . 4 ⊢ (∃y ∈ B 𝐷 = 𝐶 → 𝐷 ∈ V)
8	7	rexlimivw 2407	. . 3 ⊢ (∃x ∈ A ∃y ∈ B 𝐷 = 𝐶 → 𝐷 ∈ V)
9		eqeq1 2028	. . . 4 ⊢ (z = 𝐷 → (z = 𝐶 ↔ 𝐷 = 𝐶))
10	9	2rexbidv 2327	. . 3 ⊢ (z = 𝐷 → (∃x ∈ A ∃y ∈ B z = 𝐶 ↔ ∃x ∈ A ∃y ∈ B 𝐷 = 𝐶))
11	8, 10	elab3 2671	. 2 ⊢ (𝐷 ∈ {z ∣ ∃x ∈ A ∃y ∈ B z = 𝐶} ↔ ∃x ∈ A ∃y ∈ B 𝐷 = 𝐶)
12	3, 11	bitri 173	1 ⊢ (𝐷 ∈ ran 𝐹 ↔ ∃x ∈ A ∃y ∈ B 𝐷 = 𝐶)

Syntax hints:   ↔ wb 98   = wceq 1228   ∈ wcel 1374  {cab 2008  ∃wrex 2285  Vcvv 2535  ran crn 4273   ↦ cmpt2 5438

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-cnv 4280  df-dm 4282  df-rn 4283  df-oprab 5440  df-mpt2 5441

This theorem is referenced by: (None)