Theorem mnurndlem2 45527

Description: Lemma for mnurnd 45528. Deduction theorem input. (Contributed by Rohan Ridenour, 13-Aug-2023.)

Hypotheses

Ref	Expression
mnurndlem2.1	⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))}
mnurndlem2.2	⊢ (𝜑 → 𝑈 ∈ 𝑀)
mnurndlem2.3	⊢ (𝜑 → 𝐴 ∈ 𝑈)
mnurndlem2.4	⊢ (𝜑 → 𝐹:𝐴⟶𝑈)
mnurndlem2.5	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
mnurndlem2	⊢ (𝜑 → ran 𝐹 ∈ 𝑈)

Distinct variable groups:   𝑈,𝑘,𝑚,𝑛,𝑞,𝑝,𝑙   𝑈,𝑟,𝑘,𝑚,𝑛,𝑝,𝑙

Allowed substitution hints:   𝜑(𝑘, 𝑚, 𝑛, 𝑟, 𝑞, 𝑝, 𝑙)   𝐴(𝑘, 𝑚, 𝑛, 𝑟, 𝑞, 𝑝, 𝑙)   𝐹(𝑘, 𝑚, 𝑛, 𝑟, 𝑞, 𝑝, 𝑙)   𝑀(𝑘, 𝑚, 𝑛, 𝑟, 𝑞, 𝑝, 𝑙)

Proof of Theorem mnurndlem2

Dummy variables 𝑣 𝑎 𝑏 𝑤 𝑖 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mnurndlem2.1	. 2 ⊢ 𝑀 = {𝑘 ∣ ∀𝑙 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑘 ∧ ∀𝑚∃𝑛 ∈ 𝑘 (𝒫 𝑙 ⊆ 𝑛 ∧ ∀𝑝 ∈ 𝑙 (∃𝑞 ∈ 𝑘 (𝑝 ∈ 𝑞 ∧ 𝑞 ∈ 𝑚) → ∃𝑟 ∈ 𝑚 (𝑝 ∈ 𝑟 ∧ ∪ 𝑟 ⊆ 𝑛))))}
2		mnurndlem2.2	. 2 ⊢ (𝜑 → 𝑈 ∈ 𝑀)
3		mnurndlem2.3	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
4	2	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑈 ∈ 𝑀)
5	3	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐴 ∈ 𝑈)
6		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
7	1, 4, 5, 6	mnutrcld 45524	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝑈)
8		mnurndlem2.4	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝐴⟶𝑈)
9	8	ffvelcdmda 7116	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ 𝑈)
10	1, 4, 9, 5	mnuprd 45521	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → {(𝐹‘𝑎), 𝐴} ∈ 𝑈)
11	1, 4, 7, 10	mnuprd 45521	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → {𝑎, {(𝐹‘𝑎), 𝐴}} ∈ 𝑈)
12	11	ralrimiva 3158	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 {𝑎, {(𝐹‘𝑎), 𝐴}} ∈ 𝑈)
13		eqid 2763	. . . . . 6 ⊢ (𝑎 ∈ 𝐴 ↦ {𝑎, {(𝐹‘𝑎), 𝐴}}) = (𝑎 ∈ 𝐴 ↦ {𝑎, {(𝐹‘𝑎), 𝐴}})
14	13	rnmptss 7155	. . . . 5 ⊢ (∀𝑎 ∈ 𝐴 {𝑎, {(𝐹‘𝑎), 𝐴}} ∈ 𝑈 → ran (𝑎 ∈ 𝐴 ↦ {𝑎, {(𝐹‘𝑎), 𝐴}}) ⊆ 𝑈)
15	12, 14	syl 18	. . . 4 ⊢ (𝜑 → ran (𝑎 ∈ 𝐴 ↦ {𝑎, {(𝐹‘𝑎), 𝐴}}) ⊆ 𝑈)
16	1, 2, 3, 15	mnuop3d 45516	. . 3 ⊢ (𝜑 → ∃𝑤 ∈ 𝑈 ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ ran (𝑎 ∈ 𝐴 ↦ {𝑎, {(𝐹‘𝑎), 𝐴}})𝑖 ∈ 𝑣 → ∃𝑢 ∈ ran (𝑎 ∈ 𝐴 ↦ {𝑎, {(𝐹‘𝑎), 𝐴}})(𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))
17		simprl 783	. . . 4 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑈 ∧ ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ ran (𝑎 ∈ 𝐴 ↦ {𝑎, {(𝐹‘𝑎), 𝐴}})𝑖 ∈ 𝑣 → ∃𝑢 ∈ ran (𝑎 ∈ 𝐴 ↦ {𝑎, {(𝐹‘𝑎), 𝐴}})(𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))) → 𝑤 ∈ 𝑈)
18		sseq2 3939	. . . . 5 ⊢ (𝑏 = 𝑤 → (ran 𝐹 ⊆ 𝑏 ↔ ran 𝐹 ⊆ 𝑤))
19	18	adantl 487	. . . 4 ⊢ (((𝜑 ∧ (𝑤 ∈ 𝑈 ∧ ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ ran (𝑎 ∈ 𝐴 ↦ {𝑎, {(𝐹‘𝑎), 𝐴}})𝑖 ∈ 𝑣 → ∃𝑢 ∈ ran (𝑎 ∈ 𝐴 ↦ {𝑎, {(𝐹‘𝑎), 𝐴}})(𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))) ∧ 𝑏 = 𝑤) → (ran 𝐹 ⊆ 𝑏 ↔ ran 𝐹 ⊆ 𝑤))
20	8	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑈 ∧ ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ ran (𝑎 ∈ 𝐴 ↦ {𝑎, {(𝐹‘𝑎), 𝐴}})𝑖 ∈ 𝑣 → ∃𝑢 ∈ ran (𝑎 ∈ 𝐴 ↦ {𝑎, {(𝐹‘𝑎), 𝐴}})(𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))) → 𝐹:𝐴⟶𝑈)
21		mnurndlem2.5	. . . . 5 ⊢ 𝐴 ∈ V
22		simprr 785	. . . . 5 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑈 ∧ ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ ran (𝑎 ∈ 𝐴 ↦ {𝑎, {(𝐹‘𝑎), 𝐴}})𝑖 ∈ 𝑣 → ∃𝑢 ∈ ran (𝑎 ∈ 𝐴 ↦ {𝑎, {(𝐹‘𝑎), 𝐴}})(𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))) → ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ ran (𝑎 ∈ 𝐴 ↦ {𝑎, {(𝐹‘𝑎), 𝐴}})𝑖 ∈ 𝑣 → ∃𝑢 ∈ ran (𝑎 ∈ 𝐴 ↦ {𝑎, {(𝐹‘𝑎), 𝐴}})(𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))
23	20, 21, 22	mnurndlem1 45526	. . . 4 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑈 ∧ ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ ran (𝑎 ∈ 𝐴 ↦ {𝑎, {(𝐹‘𝑎), 𝐴}})𝑖 ∈ 𝑣 → ∃𝑢 ∈ ran (𝑎 ∈ 𝐴 ↦ {𝑎, {(𝐹‘𝑎), 𝐴}})(𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))) → ran 𝐹 ⊆ 𝑤)
24	17, 19, 23	rspcedvd 3582	. . 3 ⊢ ((𝜑 ∧ (𝑤 ∈ 𝑈 ∧ ∀𝑖 ∈ 𝐴 (∃𝑣 ∈ ran (𝑎 ∈ 𝐴 ↦ {𝑎, {(𝐹‘𝑎), 𝐴}})𝑖 ∈ 𝑣 → ∃𝑢 ∈ ran (𝑎 ∈ 𝐴 ↦ {𝑎, {(𝐹‘𝑎), 𝐴}})(𝑖 ∈ 𝑢 ∧ ∪ 𝑢 ⊆ 𝑤)))) → ∃𝑏 ∈ 𝑈 ran 𝐹 ⊆ 𝑏)
25	16, 24	rexlimddv 3173	. 2 ⊢ (𝜑 → ∃𝑏 ∈ 𝑈 ran 𝐹 ⊆ 𝑏)
26	1, 2, 25	mnuss2d 45509	1 ⊢ (𝜑 → ran 𝐹 ∈ 𝑈)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900  𝒫 cpw 4560  {cpr 4590  ∪ cuni 4872   ↦ cmpt 5192  ran crn 5659  ⟶wf 6562  ‘cfv 6566

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 5255  ax-nul 5266  ax-pr 5398  ax-reg 9623

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-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-eprel 5558  df-fr 5611  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574

This theorem is used by:  mnurnd  45528