Theorem rn1st 45394

Description: The range of a function with a first-countable domain is itself first-countable. This is a variation of 1stcrestlem 23368, with a not-free hypothesis replacing a disjoint variable constraint. (Contributed by Glauco Siliprandi, 24-Jan-2025.)

Hypothesis

Ref	Expression
rn1st.1	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
rn1st	⊢ (𝐵 ≼ ω → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω)

Proof of Theorem rn1st

Step	Hyp	Ref	Expression
1		ordom 7812	. . . . . 6 ⊢ Ord ω
2		reldom 8881	. . . . . . . 8 ⊢ Rel ≼
3	2	brrelex2i 5676	. . . . . . 7 ⊢ (𝐵 ≼ ω → ω ∈ V)
4		elong 6319	. . . . . . 7 ⊢ (ω ∈ V → (ω ∈ On ↔ Ord ω))
5	3, 4	syl 17	. . . . . 6 ⊢ (𝐵 ≼ ω → (ω ∈ On ↔ Ord ω))
6	1, 5	mpbiri 258	. . . . 5 ⊢ (𝐵 ≼ ω → ω ∈ On)
7		ondomen 9935	. . . . 5 ⊢ ((ω ∈ On ∧ 𝐵 ≼ ω) → 𝐵 ∈ dom card)
8	6, 7	mpancom 688	. . . 4 ⊢ (𝐵 ≼ ω → 𝐵 ∈ dom card)
9		rn1st.1	. . . . 5 ⊢ Ⅎ𝑥𝐵
10		eqid 2733	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)
11	9, 10	dmmptssf 45353	. . . 4 ⊢ dom (𝑥 ∈ 𝐵 ↦ 𝐶) ⊆ 𝐵
12		ssnum 9937	. . . 4 ⊢ ((𝐵 ∈ dom card ∧ dom (𝑥 ∈ 𝐵 ↦ 𝐶) ⊆ 𝐵) → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ dom card)
13	8, 11, 12	sylancl 586	. . 3 ⊢ (𝐵 ≼ ω → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ dom card)
14		funmpt 6524	. . . 4 ⊢ Fun (𝑥 ∈ 𝐵 ↦ 𝐶)
15		funforn 6747	. . . 4 ⊢ (Fun (𝑥 ∈ 𝐵 ↦ 𝐶) ↔ (𝑥 ∈ 𝐵 ↦ 𝐶):dom (𝑥 ∈ 𝐵 ↦ 𝐶)–onto→ran (𝑥 ∈ 𝐵 ↦ 𝐶))
16	14, 15	mpbi 230	. . 3 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶):dom (𝑥 ∈ 𝐵 ↦ 𝐶)–onto→ran (𝑥 ∈ 𝐵 ↦ 𝐶)
17		fodomnum 9955	. . 3 ⊢ (dom (𝑥 ∈ 𝐵 ↦ 𝐶) ∈ dom card → ((𝑥 ∈ 𝐵 ↦ 𝐶):dom (𝑥 ∈ 𝐵 ↦ 𝐶)–onto→ran (𝑥 ∈ 𝐵 ↦ 𝐶) → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ dom (𝑥 ∈ 𝐵 ↦ 𝐶)))
18	13, 16, 17	mpisyl 21	. 2 ⊢ (𝐵 ≼ ω → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ dom (𝑥 ∈ 𝐵 ↦ 𝐶))
19		ctex 8892	. . . 4 ⊢ (𝐵 ≼ ω → 𝐵 ∈ V)
20		ssdomg 8929	. . . 4 ⊢ (𝐵 ∈ V → (dom (𝑥 ∈ 𝐵 ↦ 𝐶) ⊆ 𝐵 → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ 𝐵))
21	19, 11, 20	mpisyl 21	. . 3 ⊢ (𝐵 ≼ ω → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ 𝐵)
22		domtr 8936	. . 3 ⊢ ((dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ 𝐵 ∧ 𝐵 ≼ ω) → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω)
23	21, 22	mpancom 688	. 2 ⊢ (𝐵 ≼ ω → dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω)
24		domtr 8936	. 2 ⊢ ((ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ dom (𝑥 ∈ 𝐵 ↦ 𝐶) ∧ dom (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω) → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω)
25	18, 23, 24	syl2anc 584	1 ⊢ (𝐵 ≼ ω → ran (𝑥 ∈ 𝐵 ↦ 𝐶) ≼ ω)

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2113  Ⅎwnfc 2880  Vcvv 3437   ⊆ wss 3898   class class class wbr 5093   ↦ cmpt 5174  dom cdm 5619  ran crn 5620  Ord word 6310  Oncon0 6311  Fun wfun 6480  –onto→wfo 6484  ωcom 7802   ≼ cdom 8873  cardccrd 9835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-er 8628  df-map 8758  df-en 8876  df-dom 8877  df-card 9839  df-acn 9842

This theorem is referenced by:  saliunclf  46444