Theorem mptssid 39967

Description: The mapping operation expressed with its actual domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
mptssid.1	⊢ Ⅎ𝑥𝐴
mptssid.2	⊢ 𝐶 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}

Assertion

Ref	Expression
mptssid	⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐵)

Proof of Theorem mptssid

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 474	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 ∈ 𝐴)
2		eqvisset 3351	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐵 → 𝐵 ∈ V)
3	2	adantl 473	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝐵 ∈ V)
4	1, 3	jca 555	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V))
5		rabid 3254	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ V))
6	4, 5	sylibr 224	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V})
7		mptssid.2	. . . . . . . 8 ⊢ 𝐶 = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V}
8	6, 7	syl6eleqr 2850	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑥 ∈ 𝐶)
9		simpr 479	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
10	8, 9	jca 555	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵))
11		mptssid.1	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐴
12	11	ssrab2f 39817	. . . . . . . . . 10 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ 𝐴
13	7, 12	eqsstri 3776	. . . . . . . . 9 ⊢ 𝐶 ⊆ 𝐴
14		id 22	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐶)
15	13, 14	sseldi 3742	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴)
16	15	adantr 472	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → 𝑥 ∈ 𝐴)
17		simpr 479	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → 𝑦 = 𝐵)
18	16, 17	jca 555	. . . . . 6 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵) → (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))
19	10, 18	impbii 199	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵))
20	19	ax-gen 1871	. . . 4 ⊢ ∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵))
21	20	ax-gen 1871	. . 3 ⊢ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵))
22		eqopab2b 5155	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵)} ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵)))
23	21, 22	mpbir 221	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵)}
24		df-mpt 4882	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
25		df-mpt 4882	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐵)}
26	23, 24, 25	3eqtr4i 2792	1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐵)

Syntax hints:   ↔ wb 196   ∧ wa 383  ∀wal 1630   = wceq 1632   ∈ wcel 2139  Ⅎwnfc 2889  {crab 3054  Vcvv 3340  {copab 4864   ↦ cmpt 4881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-opab 4865  df-mpt 4882

This theorem is referenced by:  limsupequzmpt2  40471  liminfequzmpt2  40544