Theorem mnurndlem2 41356

Description: Lemma for mnurnd 41357. 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 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑈 ∈ 𝑀)
5	3	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐴 ∈ 𝑈)
6		simpr 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
7	1, 4, 5, 6	mnutrcld 41353	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝑈)
8		mnurndlem2.4	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝐴⟶𝑈)
9	8	ffvelrnda 6843	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ 𝑈)
10	1, 4, 9, 5	mnuprd 41350	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → {(𝐹‘𝑎), 𝐴} ∈ 𝑈)
11	1, 4, 7, 10	mnuprd 41350	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → {𝑎, {(𝐹‘𝑎), 𝐴}} ∈ 𝑈)
12	11	ralrimiva 3114	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 {𝑎, {(𝐹‘𝑎), 𝐴}} ∈ 𝑈)
13		eqid 2759	. . . . . 6 ⊢ (𝑎 ∈ 𝐴 ↦ {𝑎, {(𝐹‘𝑎), 𝐴}}) = (𝑎 ∈ 𝐴 ↦ {𝑎, {(𝐹‘𝑎), 𝐴}})
14	13	rnmptss 6878	. . . . 5 ⊢ (∀𝑎 ∈ 𝐴 {𝑎, {(𝐹‘𝑎), 𝐴}} ∈ 𝑈 → ran (𝑎 ∈ 𝐴 ↦ {𝑎, {(𝐹‘𝑎), 𝐴}}) ⊆ 𝑈)
15	12, 14	syl 17	. . . 4 ⊢ (𝜑 → ran (𝑎 ∈ 𝐴 ↦ {𝑎, {(𝐹‘𝑎), 𝐴}}) ⊆ 𝑈)
16	1, 2, 3, 15	mnuop3d 41345	. . 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 771	. . . 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 3919	. . . . 5 ⊢ (𝑏 = 𝑤 → (ran 𝐹 ⊆ 𝑏 ↔ ran 𝐹 ⊆ 𝑤))
19	18	adantl 486	. . . 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 485	. . . . 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 773	. . . . 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 41355	. . . 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 3545	. . 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 3216	. 2 ⊢ (𝜑 → ∃𝑏 ∈ 𝑈 ran 𝐹 ⊆ 𝑏)
26	1, 2, 25	mnuss2d 41338	1 ⊢ (𝜑 → ran 𝐹 ∈ 𝑈)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1537   = wceq 1539   ∈ wcel 2112  {cab 2736  ∀wral 3071  ∃wrex 3072  Vcvv 3410   ⊆ wss 3859  𝒫 cpw 4495  {cpr 4525  ∪ cuni 4799   ↦ cmpt 5113  ran crn 5526  ⟶wf 6332  ‘cfv 6336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-reg 9082

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-eprel 5436  df-fr 5484  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-fv 6344

This theorem is referenced by:  mnurnd  41357