Theorem mnurndlem2 43041

Description: Lemma for mnurnd 43042. 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 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑈 ∈ 𝑀)
5	3	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐴 ∈ 𝑈)
6		simpr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
7	1, 4, 5, 6	mnutrcld 43038	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝑈)
8		mnurndlem2.4	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝐴⟶𝑈)
9	8	ffvelcdmda 7087	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝐹‘𝑎) ∈ 𝑈)
10	1, 4, 9, 5	mnuprd 43035	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → {(𝐹‘𝑎), 𝐴} ∈ 𝑈)
11	1, 4, 7, 10	mnuprd 43035	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → {𝑎, {(𝐹‘𝑎), 𝐴}} ∈ 𝑈)
12	11	ralrimiva 3147	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 {𝑎, {(𝐹‘𝑎), 𝐴}} ∈ 𝑈)
13		eqid 2733	. . . . . 6 ⊢ (𝑎 ∈ 𝐴 ↦ {𝑎, {(𝐹‘𝑎), 𝐴}}) = (𝑎 ∈ 𝐴 ↦ {𝑎, {(𝐹‘𝑎), 𝐴}})
14	13	rnmptss 7122	. . . . 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 43030	. . 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 770	. . . 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 4009	. . . . 5 ⊢ (𝑏 = 𝑤 → (ran 𝐹 ⊆ 𝑏 ↔ ran 𝐹 ⊆ 𝑤))
19	18	adantl 483	. . . 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 482	. . . . 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 772	. . . . 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 43040	. . . 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 3615	. . 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 3162	. 2 ⊢ (𝜑 → ∃𝑏 ∈ 𝑈 ran 𝐹 ⊆ 𝑏)
26	1, 2, 25	mnuss2d 43023	1 ⊢ (𝜑 → ran 𝐹 ∈ 𝑈)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397  ∀wal 1540   = wceq 1542   ∈ wcel 2107  {cab 2710  ∀wral 3062  ∃wrex 3071  Vcvv 3475   ⊆ wss 3949  𝒫 cpw 4603  {cpr 4631  ∪ cuni 4909   ↦ cmpt 5232  ran crn 5678  ⟶wf 6540  ‘cfv 6544

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-reg 9587

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-eprel 5581  df-fr 5632  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552

This theorem is referenced by:  mnurnd  43042